#7614240

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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

1757 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)(4,8)(5,6)(9,10)(11,15)(12,23)(13,17)(14,19)(16,25)(18,28)(20,32)(21,31)(22,35)(24,29)(26,33)(27,30)(34,36)(37,40)(38,39);(2,6)(7,8)(9,11,14,21)(10,15,19,31)(12,24,26,38)(13,27,28,22)(16,32,34,40)(17,30,18,35)(20,36,37,25)(23,29,33,39);(1,3,5,4)(2,7,6,8)(9,12,13,25,14,26,28,36)(10,16,18,23,19,34,17,33)(11,20,22,24,21,37,27,38)(15,29,30,32,31,39,35,40);(10,17,19,18)(11,21)(15,30,31,35)(16,33,34,23)(20,37)(22,27)(24,38)(29,32,39,40);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=hi,e^2=f,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521436);