#7652634

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

4908 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,e^2=ghi,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, 7559830);