#7534285

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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)(11,15)(12,14)(13,16)(17,18)(19,30)(20,26)(21,27)(22,39)(23,29)(24,28)(25,34)(31,45)(32,47)(33,42)(35,46)(36,44)(37,40)(38,43)(41,48);(2,6)(7,8)(11,16)(13,15)(17,19,23,34)(18,25,29,30)(20,31,21,37)(22,35,24,33)(26,40,27,45)(28,46,39,42)(32,36,48,38)(41,44,47,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,27)(19,31,34,37)(22,38,24,36)(25,40,30,45)(28,44,39,43)(32,33,48,35)(41,46,47,42);(10,14)(13,15)(17,21,23,20)(18,27,29,26)(19,32,34,48)(22,38,24,36)(25,41,30,47)(28,44,39,43)(31,33,37,35)(40,46,45,42);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,36,21,38)(25,42,30,46)(26,43,27,44)(31,32,37,48)(40,47,45,41)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,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,e^b=ei,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, 7441481);