#506208

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	#28097
prime radical	#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|b^2=egh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,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	#72217290, #72343002
maximal normal subgroup of	#72217290, #72343002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413404);