#7741188

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

5437 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,14)(13,15)(18,19)(23,25)(24,26)(30,31);(9,11,13,12)(10,14,15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(2,5)(6,7)(10,15)(14,16)(18,19)(20,27)(24,26)(28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,15,14)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=g,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, 7648384);