#10320502

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

4723 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^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, 10227698);