#8253945

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

4804 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:

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

Polycyclic presentation:

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

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, 8161141);