#10451675

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

5084 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)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32);(9,10,12,14)(11,13,16,15)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31);(1,3)(2,6)(4,7)(5,8)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)

Polycyclic presentation:

2^9|b^2=g,d^2=g,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, 10358871);