#7584459

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1252 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fgi,d^2=i,e^2=gi,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, 7491655);