#7854698

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

4717 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,23,22)(19,26,21,24)(20,25,30,29)(27,32,28,31);(1,3,7,5)(2,4,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,22)(19,24,21,26)(20,25,30,29)(27,31,28,32);(1,4,7,6)(2,5,8,3)(17,19,23,21)(18,24,22,26)(20,28,30,27)(25,32,29,31);(1,3,7,5)(2,4,8,6)(9,11,13,12)(10,14,16,15)(17,20,23,30)(18,25,22,29)(19,27,21,28)(24,31,26,32);(1,5,7,3)(2,4,8,6)(9,11,13,12)(10,14,16,15)(17,21,23,19)(18,26,22,24)(20,28,30,27)(25,32,29,31);(1,6,7,4)(2,5,8,3)(9,12,13,11)(10,15,16,14)(17,22,23,18)(19,26,21,24)(20,29,30,25)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=gh,f^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, 7761894);