#6812044

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,d^2=f,e^2=fhi,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, 6719240);