#8755019

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

4618 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)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31);(2,6)(4,7)(10,13)(14,15)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(18,25)(19,28)(20,27)(26,30);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30);(1,4,5,7)(2,3,6,8)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24);(1,3)(2,7)(4,6)(5,8)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662215);