#9203858

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	#49380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,e^2=gi,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, 9111054);