#7645375

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

4815 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)(11,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,23)(22,24)(25,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(17,19,23,22)(18,24,21,27)(20,28,30,29)(25,31,26,32);(9,12,14,10)(11,15,13,16)(17,20)(18,25)(19,28)(21,26)(22,29)(23,30)(24,31)(27,32);(9,13,14,11)(10,15,12,16)(17,21,23,18)(19,27,22,24)(20,26,30,25)(28,32,29,31);(17,20)(18,26)(19,28)(21,25)(22,29)(23,30)(24,32)(27,31);(1,3,4,7)(2,6,5,8)(17,22,23,19)(18,27,21,24)(20,29,30,28)(25,32,26,31)

Polycyclic presentation:

2^9|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, 7552571);