#6674434

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6581630);