#9210394

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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)(11,12)(13,21)(14,17)(15,19)(16,23)(18,22)(20,24)(25,26,31,27)(28,34,30,32)(29,33,38,35)(36,40,37,39);(1,2,6,8)(3,5,7,4)(9,11,14,19)(10,15,17,12)(13,18,23,24)(16,22,21,20)(25,27,31,26)(28,34,30,32)(29,33,38,35)(36,39,37,40);(1,3,6,7)(2,4,8,5)(9,10,14,17)(11,15,19,12)(13,16,23,21)(18,22,24,20)(25,28,31,30)(26,32,27,34)(29,36,38,37)(33,39,35,40);(1,4,6,5)(2,3,8,7)(9,10)(11,15)(12,19)(13,16)(14,17)(18,22)(20,24)(21,23);(1,2)(3,4)(5,7)(6,8)(9,12,14,15)(10,11,17,19)(13,22,23,20)(16,24,21,18)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,5,6,4)(2,7,8,3)(9,13)(10,16)(11,18)(12,20)(14,23)(15,22)(17,21)(19,24)(25,30,31,28)(26,34,27,32)(29,37,38,36)(33,40,35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117590);