#7575687

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(4,8)(5,6)(10,14)(11,12)(17,18,21,25)(19,23,29,37)(20,31,32,24)(22,26,34,40)(27,41,43,35)(28,36,30,38)(33,45,46,39)(42,48,44,47);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,37)(26,36)(30,34)(31,41)(32,43)(33,42)(38,40)(39,47)(44,46)(45,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,21,32)(18,24,25,31)(19,27,29,43)(22,33,34,46)(23,35,37,41)(26,39,40,45)(28,42,30,44)(36,47,38,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,16,14,15)(19,28)(22,34)(23,36)(26,40)(27,42)(29,30)(33,46)(35,47)(37,38)(39,45)(41,48)(43,44);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(20,32)(24,31)(27,43)(33,46)(35,41)(39,45)(42,44)(47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=f,e^2=gi,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, 7482883);