#7548630

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,38)(22,28)(23,27)(24,33)(26,37)(30,43)(31,41)(32,42)(34,46)(35,40)(36,45)(39,44)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,33)(18,24,28,29)(20,35,37,30)(21,34,23,32)(25,43,26,40)(27,46,38,42)(31,39,48,36)(41,45,47,44);(10,14)(11,16)(17,20)(18,25)(19,30)(21,39)(22,37)(23,36)(24,40)(26,28)(27,45)(29,43)(31,34)(32,48)(33,35)(38,44)(41,46)(42,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,26)(19,31)(21,36)(22,37)(23,39)(24,41)(25,28)(27,45)(29,47)(30,34)(32,35)(33,48)(38,44)(40,46)(42,43);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,36,37,39)(24,42,29,46)(25,44,26,45)(30,31,35,48)(40,41,43,47)

Polycyclic presentation:

2^9|b^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,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, 7455826);