#512736

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2163
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ch,e^a=ei,d^b=di,f^b=fi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217181, #72217183, #72217215, …
maximal normal subgroup of	#72217181, #72217183, #72217215, …
maximal quotient of	C₂×#512736

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419932);