#505707

order	512
nilpotency class	4
isoclinism	isoclinic class of #500851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27662
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,15,8,16,9,14)(3,17,35,52,23,53,22,62)(4,24,36,44,27,50,26,11)(5,28,30,63,19,56,31,51)(6,32,33,49,34,48,18,13)(10,37,40,61,41,54,42,21)(12,45,46,58,47,55,38,29)(20,59,25,39,57,64,60,43);(1,3,18,54,9,22,34,61,8,23,33,37,7,35,6,21)(2,10,32,52,15,40,49,53,16,41,48,62,14,42,13,17)(4,5,20,55,26,31,60,58,27,19,57,45,36,30,25,29)(11,38,59,28,24,12,39,63,44,46,64,56,50,47,43,51);(1,4)(2,11)(3,19)(5,23)(6,25)(7,36)(8,27)(9,26)(10,38)(12,40)(13,43)(14,50)(15,24)(16,44)(17,51)(18,20)(21,58)(22,30)(28,52)(29,61)(31,35)(32,59)(33,57)(34,60)(37,55)(39,49)(41,46)(42,47)(45,54)(48,64)(53,63)(56,62);(1,5,18,55,9,31,34,58,8,19,33,45,7,30,6,29)(2,12,32,63,15,46,49,56,16,47,48,51,14,38,13,28)(3,20,54,26,22,60,61,27,23,57,37,36,35,25,21,4)(10,39,52,44,40,64,53,50,41,43,62,11,42,59,17,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=eh,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72225003, #72225011, …
maximal normal subgroup of	#72217287, #72225003, #72225011, …
maximal quotient of	C₂×#505707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412903);