#8438549

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

5854 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8345745);