#10097166

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

4746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,f^2=h,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, 10004362);