#7865032

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

5413 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772228);