#506331

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	#28188
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72320907, C₂×#506331
maximal normal subgroup of	#72217287, #72320907, C₂×#506331
maximal quotient of	C₂×#506331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413527);