#8254693

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,31,32)(27,35,39,33)(28,34,29,30)(36,40,37,38);(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,21)(22,23)(26,32)(28,29)(33,35)(36,37);(2,4)(6,7)(12,20)(16,21)(18,24)(22,23)(25,27,31,39)(26,33,32,35)(28,36,29,37)(30,40,34,38);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,29)(26,30,32,34)(27,36,39,37)(33,40,35,38);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,31,28)(26,34,32,30)(27,37,39,36)(33,38,35,40);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,34)(26,28,32,29)(27,38,39,40)(33,37,35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8161889);