#8341812

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

4756 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,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, 8249008);