#9910223

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

4605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817419);