#7654603

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

5374 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,e^2=g,f^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, 7561799);