#10324479

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

4844 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 10231675);