#7534610

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 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)(19,25)(20,26)(21,36)(22,27)(23,28)(24,29)(30,39)(31,46)(32,41)(33,42)(34,43)(35,44)(37,48)(38,45)(40,47);(2,6)(7,8)(11,16)(13,15)(17,19,24,34)(18,25,29,43)(20,30,21,37)(22,33,23,32)(26,39,36,48)(27,42,28,41)(31,35,47,38)(40,45,46,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,24,21)(18,26,29,36)(19,30,34,37)(22,38,23,35)(25,39,43,48)(27,45,28,44)(31,32,47,33)(40,42,46,41);(10,14)(13,15)(17,21,24,20)(18,26,29,36)(19,31,34,47)(22,38,23,35)(25,40,43,46)(27,44,28,45)(30,32,37,33)(39,42,48,41);(1,4,5,3)(2,7,6,8)(17,22,24,23)(18,27,29,28)(19,32,34,33)(20,35,21,38)(25,41,43,42)(26,44,36,45)(30,31,37,47)(39,46,48,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=fi,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, 7441806);