#7539970

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

1529 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)(19,25)(20,26)(21,36)(22,28)(23,27)(24,31)(29,41)(30,39)(32,48)(33,45)(34,44)(35,43)(37,40)(38,46)(42,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,31)(23,37)(25,28)(26,42)(27,39)(29,35)(33,46)(34,47)(36,40)(38,43)(41,45)(44,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,25)(20,29)(21,37)(22,31)(23,30)(24,28)(26,41)(27,39)(32,35)(33,47)(34,46)(36,40)(38,44)(42,45)(43,48);(10,14)(13,15)(17,20,22,34)(18,26,28,44)(19,29,31,46)(21,35,23,33)(24,38,25,41)(27,45,36,43)(30,47,37,32)(39,42,40,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,36)(19,30,31,37)(20,33,34,35)(24,39,25,40)(26,43,44,45)(29,32,46,47)(38,48,41,42)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,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, 7447166);