#6669870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6577066);