#7528313

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,22,27)(19,28,31,24)(20,32,35,25)(21,36,23,26)(29,39,47,45)(30,40,37,41)(33,42,46,48)(34,43,38,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,28)(20,33,35,46)(21,30,23,37)(25,42,32,48)(26,40,36,41)(29,38,47,34)(39,43,45,44);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19)(18,24)(20,29)(21,37)(22,31)(23,30)(25,39)(26,41)(27,28)(32,45)(33,34)(35,47)(36,40)(38,46)(42,44)(43,48);(10,14)(13,16)(17,20,22,35)(18,25,27,32)(19,29,31,47)(21,34,23,38)(24,39,28,45)(26,43,36,44)(30,46,37,33)(40,42,41,48);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,32,44)(29,46,47,33)(39,42,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=i,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, 7435509);