#8891371

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

4776 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8798567);