#7684583

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

5161 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,15)(14,16)(18,21)(22,24);(1,3,5,4)(2,6,7,8)(17,18)(19,22)(20,21)(23,24);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,15,16,12);(9,12,14,15)(10,11,16,13);(1,2)(3,6)(4,8)(5,7)(9,13,14,11)(10,15,16,12)(17,19,20,23)(18,22,21,24);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 7591779);