#7527634

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

1882 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,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, 7434830);