#7534582

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(17,18,23,29)(19,25,34,42)(20,27,21,26)(22,28,24,30)(31,44,36,39)(32,48,47,40)(33,41,35,43)(37,46,38,45);(2,6)(7,8)(11,16)(13,15)(17,19)(18,25)(20,36)(21,31)(22,35)(23,34)(24,33)(26,44)(27,39)(28,43)(29,42)(30,41)(32,38)(37,47)(40,46)(45,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,27)(19,31,34,36)(22,38,24,37)(25,39,42,44)(28,46,30,45)(32,35,47,33)(40,43,48,41);(10,14)(13,15)(17,21,23,20)(18,27,29,26)(19,32,34,47)(22,38,24,37)(25,40,42,48)(28,46,30,45)(31,35,36,33)(39,43,44,41);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,33,34,35)(20,37,21,38)(25,41,42,43)(26,45,27,46)(31,47,36,32)(39,48,44,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=i,e^2=fi,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, 7441778);