#10261473

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

5191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10168669);