#8341809

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

4859 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8249005);