#8968057

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

4508 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,21)(14,15)(18,22)(19,23)(20,24)(25,26)(27,30)(28,37)(29,32)(31,34)(33,36)(35,39)(38,40);(2,6)(3,8)(10,15)(16,17)(18,21)(23,24)(25,27)(26,30)(28,38)(29,35)(31,36)(32,39)(33,34)(37,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,12)(10,16,15,17)(13,20,22,19)(18,24,21,23)(25,28,31,29)(26,32,34,37)(27,35,36,38)(30,40,33,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,29,31,28)(26,32,34,37)(27,35,36,38)(30,39,33,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,30,31,33)(26,27,34,36)(28,39,29,40)(32,38,37,35);(1,4)(2,3)(5,7)(6,8)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,27)(26,33)(28,38)(29,35)(30,34)(31,36)(32,40)(37,39)

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8875253);