#7514250

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

3196 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)(9,10,14,19)(11,20,23,16)(12,24,27,17)(13,15,28,18)(21,31,25,33)(22,32,36,29)(26,34,38,30)(35,39,37,40);(1,2)(3,4)(5,6)(7,8)(10,15)(13,28)(16,29)(17,30)(18,19)(20,32)(22,36)(24,34)(26,38)(31,39)(33,40)(35,37);(6,8)(9,11,14,23)(10,16,19,20)(12,25,27,21)(13,22,28,36)(15,29,18,32)(17,33,24,31)(26,37,38,35)(30,40,34,39);(2,4)(6,8)(9,12,14,27)(10,17,19,24)(11,21,23,25)(13,26,28,38)(15,30,18,34)(16,31,20,33)(22,35,36,37)(29,39,32,40);(10,15)(13,28)(16,29)(17,30)(18,19)(20,32)(22,36)(24,34)(26,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421446);