#7563118

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
prime radical	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 7470314);