#7582318

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,46)(32,45)(33,39)(34,40)(35,48)(36,42)(37,43)(38,44)(41,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,25,28,30)(21,35,37,47)(22,32,24,34)(26,41,43,48)(27,40,29,45)(31,36,33,38)(39,42,46,44);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,19)(18,25,28,30)(21,33,37,31)(22,32,24,34)(26,39,43,46)(27,45,29,40)(35,38,47,36)(41,42,48,44);(10,14)(13,15)(17,21,23,37)(18,26,28,43)(19,31,20,33)(22,38,24,36)(25,39,30,46)(27,44,29,42)(32,47,34,35)(40,48,45,41);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,27,28,29)(19,32,20,34)(21,36,37,38)(25,40,30,45)(26,42,43,44)(31,35,33,47)(39,41,46,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7489514);