#8932583

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

4858 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)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18,23,26)(19,20,24,21)(22,25,30,32)(27,28,31,29);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19,23,24)(18,21,26,20)(22,27,30,31)(25,29,32,28);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,24,26,19)(22,29,30,28)(25,27,32,31);(2,6)(3,8)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,20)(18,24,26,19)(22,29,30,28)(25,31,32,27);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,30)(18,25,26,32)(19,27,24,31)(20,28,21,29)

Polycyclic presentation:

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

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, 8839779);