#10324283

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

4899 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:

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

Polycyclic presentation:

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

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, 10231479);