#10480697

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

5243 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)(11,12)(13,14)(15,16)(17,18)(19,22)(20,29)(21,25)(23,27)(24,26)(28,32)(30,31);(10,14)(11,15)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,20,23,21)(18,25,27,29)(19,28,26,31)(22,30,24,32);(3,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,21,23,20)(18,25,27,29)(19,28,26,31)(22,32,24,30);(1,3)(2,6)(4,7)(5,8)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,30)(29,32);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,27)(20,30)(21,32)(23,24)(25,31)(28,29)

Polycyclic presentation:

2^9|c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10387893);