#505706

order	512
nilpotency class	4
isoclinism	isoclinic class of #500850
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

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

normal subgroups only

This group has 458 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:

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

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,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217290, #72225004, #72225012, …
maximal normal subgroup of	#72217290, #72225004, #72225012, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412902);