#7649937

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

4953 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557133);