#7877239

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

5337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7784435);