#7689748

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

5250 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596944);