#6672618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579814);