#7043512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,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, 6950708);