#8443743

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

4389 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8350939);