#7519243

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

2665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=f,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=cg,d^b=dh,e^b=ei,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426439);