#7576636

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18)(19,23)(20,24)(21,26)(22,25)(27,35)(28,42)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(43,47)(44,46)(45,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,22,30)(18,23,25,37)(20,31,32,27)(21,28,34,29)(24,38,39,35)(26,42,41,36)(33,44,45,43)(40,46,48,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,39)(26,40)(28,43)(29,44)(30,31)(34,45)(36,46)(37,38)(41,48)(42,47);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(18,25)(19,28)(21,34)(23,36)(24,39)(27,43)(29,30)(31,44)(33,45)(35,46)(37,42)(38,47);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(20,32)(24,39)(27,31)(33,45)(35,38)(40,48)(43,44)(46,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dh,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, 7483832);