#7533521

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7440717);