#7530073

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1601 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)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,25)(21,26)(22,37)(23,28)(24,27)(30,46)(31,40)(32,39)(33,41)(34,48)(35,45)(36,44)(38,43)(42,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,23)(21,34)(22,31)(24,33)(26,42)(27,40)(28,29)(30,35)(32,38)(36,47)(37,41)(39,45)(43,46)(44,48);(10,14)(11,15)(17,20,23,19)(18,25,28,29)(21,32,36,30)(22,31,24,33)(26,39,44,46)(27,41,37,40)(34,35,47,38)(42,43,48,45);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,36)(24,38)(25,39)(27,43)(28,44)(29,46)(31,47)(33,34)(37,45)(40,42)(41,48);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,27,28,37)(19,31,20,33)(21,35,36,38)(25,40,29,41)(26,43,44,45)(30,47,32,34)(39,42,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fi,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, 7437269);