#10320515

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

4911 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)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32);(1,2)(3,7)(4,5)(6,8)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(18,23)(22,25)(24,28)(31,32);(2,5)(6,7)(11,15)(13,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)

Polycyclic presentation:

2^9|a^2=gh,b^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, 10227711);