#9105300

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

4969 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)(11,15)(12,16)(13,14)(17,18)(19,20)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(17,19)(18,21)(20,27)(22,23)(24,28)(25,29)(26,30)(31,32);(2,3)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20,24,21)(18,19,27,28)(22,29,31,26)(23,25,32,30);(2,3)(4,8)(9,12)(10,11)(13,15)(14,16)(17,21,24,20)(18,19,27,28)(22,26,31,29)(23,25,32,30);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,22)(18,25)(19,23)(20,26)(21,29)(24,31)(27,30)(28,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,23)(18,26)(19,22)(20,25)(21,30)(24,32)(27,29)(28,31)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9012496);