#8891265

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

4930 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798461);