#500850

order	512
nilpotency class	4
isoclinism	isoclinic class of #500850
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24267
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

953 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,13)(4,19)(5,23)(6,26)(7,11)(8,16)(9,20)(10,14)(12,24)(15,40)(17,33)(18,46)(21,52)(22,48)(25,53)(27,45)(28,55)(29,57)(30,51)(31,58)(32,39)(34,41)(35,37)(36,44)(38,47)(42,56)(43,54)(49,50)(59,63)(60,64)(61,62);(1,3,14,37,6,18,25,45,7,17,39,57,24,44,5,16)(2,8,23,36,12,29,32,33,11,27,53,46,26,35,10,13)(4,15,38,61,21,43,51,63,22,42,58,64,50,55,20,41)(9,28,49,60,31,56,48,59,30,54,52,62,47,40,19,34);(8,27)(13,33)(28,54)(29,35)(34,59)(36,46)(40,56)(60,62);(1,4,14,38,6,21,25,51,7,22,39,58,24,50,5,20)(2,9,23,49,12,31,32,48,11,30,53,52,26,47,10,19)(3,15,37,61,18,43,45,63,17,42,57,64,44,55,16,41)(8,28,36,60,29,56,33,59,27,54,46,62,35,40,13,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72224993, #72225013, …
maximal normal subgroup of	#72217287, #72224993, #72225013, …
maximal quotient of	C₂×#500850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408046);