#7519198

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

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, 7426394);