#9712331

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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9619527);