#10320516

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

5126 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,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, 10227712);