#7654927

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

5062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562123);