#7563274

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2771 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:

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

Polycyclic presentation:

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

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, 7470470);