#8817744

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

5715 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)(9,10,13,15)(11,12,16,14)(17,18)(19,24)(20,23)(21,26)(22,25)(27,31)(28,29)(30,32);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(19,26)(21,24)(27,28)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,19,22,26)(18,21,25,24)(20,28,30,27)(23,31,32,29);(9,12)(10,11)(13,14)(15,16)(17,20,22,30)(18,23,25,32)(19,27,26,28)(21,29,24,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,27)(21,25)(22,26)(23,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,24)(18,19,25,26)(20,29,30,31)(23,27,32,28)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8724940);