#7649165

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

5015 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)(19,25)(20,23)(21,26)(22,24)(27,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,16,15)(17,19,22,20)(18,23,24,25)(21,27,29,28)(26,30,31,32);(9,10)(11,13)(12,14)(15,16)(18,24)(23,25)(26,31)(30,32);(9,11,12,16)(10,13,14,15)(17,20,22,19)(18,25,24,23)(21,27,29,28)(26,30,31,32);(1,3,4,7)(2,6,5,8)(17,21,22,29)(18,26,24,31)(19,27,20,28)(23,30,25,32);(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=g,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, 7556361);