#8003808

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

4726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7911004);