#7988191

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895387);