#7743199

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

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=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, 7650395);