#506330

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	#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

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

normal subgroups only

This group has 447 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)(8,28,30,31)(15,42,44,38)(16,45,46,41)(18,26,48,53)(29,55,56,54)(43,58,62,61)(47,63,64,50);(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)(4,9,20,33,19,36,49,60,22,34,52,59,21,32,51,57)(16,43,47,54,45,61,63,56,46,62,64,55,41,58,50,29);(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)(19,41)(20,50)(21,45)(22,46)(27,53)(32,54)(33,58)(34,56)(35,48)(36,55)(37,44)(40,42)(43,57)(47,51)(49,64)(52,63)(59,61)(60,62);(1,4,5,21,7,22,25,19)(2,9,12,32,13,34,10,36)(3,16,23,45,17,46,14,41)(6,20,24,51,27,52,35,49)(8,29,28,54,30,56,31,55)(11,33,39,57,40,59,37,60)(15,43,42,61,44,62,38,58)(18,47,53,63,48,64,26,50)

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=de,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

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217290, #72320748, #72342968, …
maximal normal subgroup of	#72217290, #72320748, #72342968, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413526);