#7887136

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

4761 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)(3,6)(4,7)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,28,26,30)(21,29,32,31);(1,4)(2,3)(5,6)(7,8)(18,25)(20,26)(24,27)(28,30);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20,23,26)(18,21,25,32)(19,28,22,30)(24,29,27,31);(1,3)(2,4)(5,7)(6,8)(17,21)(18,26)(19,29)(20,25)(22,31)(23,32)(24,30)(27,28);(1,4)(2,7)(3,8)(5,6)(9,12,13,11)(10,15,16,14)(17,22,23,19)(18,27,25,24)(20,30,26,28)(21,31,32,29)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=ghi,f^2=ghi,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, 7794332);