#7516526

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=fi,g^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,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423722);