#10096974

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

4671 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=g,f^2=gh,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, 10004170);