#7645376

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

4999 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,f^2=g,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, 7552572);