#7883013

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

4811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,12)(11,15,16,14)(17,18,24,30)(19,25,21,27)(20,28,22,26)(23,29,32,31);(1,3,5,6)(2,4,7,8)(9,11,13,16)(10,14,12,15)(17,19,24,21)(18,25,30,27)(20,31,22,29)(23,26,32,28);(1,2)(3,8)(4,6)(5,7)(9,10,13,12)(11,14,16,15)(17,20,24,22)(18,26,30,28)(19,29,21,31)(23,27,32,25);(17,21,24,19)(18,27,30,25)(20,29,22,31)(23,28,32,26);(1,2)(3,4)(5,7)(6,8)(17,22,24,20)(18,28,30,26)(19,29,21,31)(23,27,32,25);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,23,24,32)(18,29,30,31)(19,26,21,28)(20,27,22,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=h,f^2=gi,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, 7790209);