#7051001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,29)(28,33)(30,32)(31,34)(35,39)(36,40)(37,38);(1,2,5,6)(3,8,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(27,34)(28,36)(38,39);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,26,30,34,31,32)(28,37,39,40,36,35,38,33);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,35)(29,39)(30,36)(31,38)(32,40)(34,37);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,34,32,27)(28,38,36,39)(33,35,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6958197);