#7694935

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

5158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602131);