#7514305

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3177 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421501);