#7654602

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

6124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561798);