#7531945

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	#45385	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(17,18,22,23)(19,24,29,38)(20,30,33,25)(21,34,37,26)(27,41,31,39)(28,35,46,40)(32,42,36,43)(44,47,45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,23,38)(20,31,33,27)(21,35,37,40)(25,41,30,39)(26,28,34,46)(32,47,36,48)(42,45,43,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,27)(21,36)(22,33)(23,30)(24,39)(26,43)(28,45)(29,31)(32,37)(34,42)(35,47)(38,41)(40,48)(44,46);(10,14)(13,15)(17,21,22,37)(18,26,23,34)(19,28,29,46)(20,32,33,36)(24,40,38,35)(25,42,30,43)(27,44,31,45)(39,47,41,48);(2,5)(6,8)(10,14)(13,15)(20,33)(25,30)(27,31)(32,36)(39,41)(42,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7439141);