#10087047

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

4909 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)(3,4)(5,8)(6,7)(17,18,21,24)(19,22,27,25)(20,23,29,32)(26,30,28,31);(1,3,5,7)(2,4,8,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15);(1,3,5,7)(2,4,8,6)(9,10)(11,13)(12,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,4)(2,7)(3,8)(5,6)(9,10)(11,13)(12,15)(14,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=g,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, 9994243);