#8599012

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

5704 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506208);