#8612816

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

4991 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=hi,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, 8520012);