#7518583

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
prime radical	#60433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,27)(14,19)(15,21)(16,23)(17,26)(18,29)(20,30)(22,37)(24,31)(25,34)(28,36)(32,35)(33,39)(38,40);(2,5)(6,7)(9,11,14,23)(10,16,19,12)(13,22,28,38)(15,17,30,24)(18,25,35,33)(20,26,21,31)(27,40,36,37)(29,39,32,34);(9,12)(10,17)(11,21)(13,25)(14,16)(15,31)(18,34)(19,24)(20,23)(22,27)(26,30)(28,33)(29,40)(32,37)(35,39)(36,38);(9,13,14,28)(10,18,19,35)(11,22,23,38)(12,25,16,33)(15,32,30,29)(17,34,24,39)(20,36,21,27)(26,40,31,37);(1,3,4,8)(2,6,5,7)(10,19)(11,23)(17,24)(18,35)(20,21)(22,38)(27,36)(34,39)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^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,e^b=ei,g^b=gi,g^c=gi,h^c=hi,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, 7425779);