#10106217

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

5111 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)(3,4)(5,7)(6,8)(10,14)(11,15)(17,18,22,25)(19,23,29,32)(20,21,30,24)(26,31,27,28);(1,3,6,5)(2,7,8,4)(9,10)(11,12)(13,14)(15,16)(17,19,22,29)(18,23,25,32)(20,27,30,26)(21,28,24,31);(9,10)(11,16)(12,15)(13,14)(17,20,22,30)(18,24,25,21)(19,26,29,27)(23,28,32,31);(1,4,6,7)(2,5,8,3)(9,11,13,15)(10,12,14,16)(17,20)(18,21)(19,27)(22,30)(23,28)(24,25)(26,29)(31,32);(1,5,6,3)(2,7,8,4)(9,12,13,16)(10,15,14,11)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,20)(19,28)(22,24)(23,27)(25,30)(26,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^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, 10013413);