#8097811

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

4944 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8005007);