#7584780

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,31)(21,34)(22,24)(23,27)(26,33)(28,43)(29,40)(30,37)(32,41)(35,39)(36,42)(38,46)(44,47)(45,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(18,24)(20,32)(21,23)(25,37)(26,41)(28,44)(29,35)(31,42)(33,36)(38,47)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,25)(20,28)(21,35)(22,30)(23,29)(24,37)(26,38)(27,40)(31,43)(32,44)(33,46)(34,39)(36,45)(41,47)(42,48);(10,14)(13,15)(17,20,22,33)(18,26,24,31)(19,28,30,46)(21,32,23,36)(25,38,37,43)(27,41,34,42)(29,45,35,44)(39,48,40,47);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,24,34)(19,29,30,35)(20,32,33,36)(25,39,37,40)(26,41,31,42)(28,45,46,44)(38,48,43,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=i,e^2=fg,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, 7491976);