#7677407

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

5517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7584603);