#7658250

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

5362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565446);