#7528989

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,35)(21,26)(22,39)(23,28)(24,27)(25,33)(30,40)(31,47)(32,42)(34,46)(36,43)(37,44)(38,45)(41,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,33)(18,25,28,29)(20,36,21,30)(22,34,24,32)(26,43,35,40)(27,46,39,42)(31,37,48,38)(41,44,47,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,21)(18,26,28,35)(19,30,33,36)(22,38,24,37)(25,40,29,43)(27,45,39,44)(31,34,48,32)(41,46,47,42);(10,14)(13,16)(17,21,23,20)(18,26,28,35)(19,31,33,48)(22,38,24,37)(25,41,29,47)(27,44,39,45)(30,34,36,32)(40,46,43,42);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,27,28,39)(19,32,33,34)(20,37,21,38)(25,42,29,46)(26,44,35,45)(30,48,36,31)(40,47,43,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7436185);