#8750823

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658019);