#10089645

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

5115 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)(3,7)(4,5)(6,8)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(18,25)(19,28)(20,30)(27,32);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,15)(14,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996841);