#8807310

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

5514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(9,10,14,13)(11,12,16,15)(17,18,21,22)(19,20,23,24);(1,2)(3,7)(4,5)(6,8)(10,13)(11,16)(18,22)(19,23);(2,5)(6,7)(9,11,14,16)(10,15,13,12)(17,19)(18,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,12,16,15);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,16)(14,15)

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714506);