#8651455

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

5280 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)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31);(1,2)(3,5)(4,6)(7,8)(10,12)(15,16)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32);(2,3)(6,7)(9,11,14,13)(10,15,12,16)(19,26)(22,25)(27,32)(30,31);(4,8)(6,7)(9,10,14,12)(11,15,13,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=hi,f^2=gi,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, 8558651);