#7534605

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,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, 7441801);