#7563357

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
prime radical	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2779 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:

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

Polycyclic presentation:

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

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, 7470553);