#7688171

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

5773 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=h,e^2=ghi,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, 7595367);