#7539982

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

1524 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)(17,18,23,28)(19,24,31,39)(20,25,35,43)(21,26,22,27)(29,37,45,47)(30,38,36,44)(32,40,46,48)(33,41,34,42);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,36)(23,31)(25,40)(26,38)(27,44)(28,39)(29,34)(33,45)(35,46)(37,42)(41,47)(43,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,37)(26,44)(27,38)(28,39)(32,34)(33,46)(35,45)(40,42)(41,48)(43,47);(10,14)(13,15)(17,20,23,35)(18,25,28,43)(19,29,31,45)(21,34,22,33)(24,37,39,47)(26,42,27,41)(30,46,36,32)(38,48,44,40);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,33,35,34)(24,38,39,44)(25,41,43,42)(29,32,45,46)(37,40,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=i,e^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,e^c=ei,e^d=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, 7447178);