#7768067

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

4828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7675263);