#7646940

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

5616 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7554136);