#10368668

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

4717 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)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,21,24,26)(22,25,31,28)(27,29,30,32);(9,11,13,16)(10,14,12,15)(17,19,23,24)(18,21,20,26)(22,30,31,27)(25,32,28,29);(1,2)(3,6)(4,5)(7,8)(10,12)(14,15)(17,18,23,20)(19,26,24,21)(22,25,31,28)(27,32,30,29);(3,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,18)(19,21,24,26)(22,28,31,25)(27,29,30,32);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,20)(22,29)(23,26)(25,30)(27,28)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,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, 10275864);