#8600254

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

4675 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gh,e^2=h,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, 8507450);