#10095066

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

4998 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10002262);