#10087731

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

4989 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^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, 9994927);