#10408485

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

4801 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10315681);