#7560693

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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

1769 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)(11,17)(12,24)(13,27)(14,19)(15,30)(16,22)(18,29)(20,25)(21,35)(23,32)(26,31)(28,40)(33,37)(34,38)(36,39);(2,5)(6,8)(9,11,14,22)(10,16,19,17)(12,25,23,15)(13,21,28,36)(18,33,34,31)(20,24,30,32)(26,38,37,29)(27,39,40,35);(1,3,4,7)(2,6,5,8)(9,12,15,11,14,23,25,22)(10,17,20,24,19,16,30,32)(13,26,29,21,28,37,38,36)(18,33,27,35,34,31,40,39);(9,13,25,38,14,28,15,29)(10,18,30,40,19,34,20,27)(11,21,12,26,22,36,23,37)(16,31,24,35,17,33,32,39);(2,5)(6,8)(10,19)(16,17)(18,34)(20,30)(24,32)(27,40)(31,33)(35,39)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=gi,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467889);