#7518282

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514160
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(11,19)(12,24)(13,15)(14,28)(16,23)(17,26)(18,27)(20,30)(21,32)(22,33)(25,34)(29,31)(35,36)(37,40)(38,39);(2,5)(6,7)(10,15)(11,20)(14,27)(16,19)(21,37)(22,23)(24,34)(26,29)(30,33)(32,35)(36,38)(39,40);(9,11)(10,16)(12,21)(13,22)(14,20)(15,30)(17,32)(18,33)(19,28)(23,27)(24,35)(25,38)(26,37)(29,36)(31,39)(34,40);(9,12,13,25)(10,17,15,31)(11,21,22,38)(14,29,27,26)(16,32,30,39)(18,34,28,24)(19,35,33,40)(20,36,23,37);(1,3,4,8)(2,6,5,7)(10,15)(16,30)(17,31)(18,28)(19,33)(24,34)(32,39)(35,40)

Polycyclic presentation:

2^9|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=cgi,d^b=dh,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, 7425478);