#7542063

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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

1688 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)(4,8)(5,7)(9,10)(11,22)(12,17)(13,28)(14,20)(15,21)(16,24)(18,30)(19,26)(23,37)(25,32)(27,31)(29,35)(33,39)(34,40)(36,38);(1,3,5,8)(2,6,7,4)(9,11,14,24,15,12,26,25)(10,16,19,22,21,32,20,17)(13,23,30,38,29,27,40,39)(18,31,28,36,34,37,35,33);(1,4,5,6)(2,3,7,8)(13,29)(18,34)(23,27)(28,35)(30,40)(31,37)(33,36)(38,39);(2,7)(3,8)(9,12)(10,17)(11,15)(13,27)(14,25)(16,19)(18,33)(20,32)(21,22)(23,29)(24,26)(28,31)(30,39)(34,36)(35,37)(38,40);(9,13,26,40,15,29,14,30)(10,18,20,35,21,34,19,28)(11,23,25,39,12,27,24,38)(16,31,17,33,32,37,22,36)

Polycyclic presentation:

2^9|b^2=fg,c^2=g,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,e^c=ei,h^d=hi

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, 7449259);