#7593014

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gi,d^2=g,e^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

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, 7500210);