#10366271

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

4589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=i,e^2=h,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, 10273467);