#8093657

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

5457 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8000853);