#8612916

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

4747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^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, 8520112);