#7514049

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

3187 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)(11,22)(12,25)(13,18)(14,19)(15,21)(16,24)(17,27)(20,30)(23,36)(26,35)(28,33)(29,32)(31,39)(34,40)(37,38);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,21,32)(12,23,27,28)(14,29,30,24)(17,31,25,38)(18,19,22,20)(26,39,40,37)(33,35,36,34);(6,8)(9,12,15,27)(10,17,21,25)(11,23,13,28)(14,26,30,40)(16,31,32,38)(18,36,22,33)(19,34,20,35)(24,37,29,39);(2,4)(6,8)(9,13,15,11)(10,18,21,22)(12,23,27,28)(14,29,30,24)(16,19,32,20)(17,33,25,36)(26,37,40,39)(31,35,38,34);(10,19)(14,30)(16,22)(17,34)(18,32)(20,21)(24,29)(25,35)(26,40)(31,33)(36,38)(37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421245);