#7517625

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
prime radical	#61198	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,e^2=i,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,e^c=ei,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, 7424821);