#7987307

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

4761 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)(3,5)(4,6)(7,8)(10,14)(15,16)(17,18,24,22)(19,20,25,21)(23,26,32,30)(27,28,31,29);(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,22)(23,31)(24,25)(26,28)(27,32)(29,30);(2,3)(6,7)(17,20,24,21)(18,25,22,19)(23,28,32,29)(26,31,30,27);(1,2,5,3)(4,6,8,7)(17,21,24,20)(18,25,22,19)(23,29,32,28)(26,31,30,27);(1,3,5,2)(4,6,8,7)(9,11,13,12)(10,15,14,16)(17,22,24,18)(19,20,25,21)(23,26,32,30)(27,29,31,28);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,32)(25,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=i,e^2=g,f^2=ghi,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, 7894503);