#7526929

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,31)(21,26)(22,27)(24,29)(25,33)(28,40)(30,39)(32,48)(34,45)(35,44)(36,41)(37,43)(38,46)(42,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,29)(18,23,26,24)(20,32,33,47)(22,36,35,30)(25,42,31,48)(27,41,44,39)(28,37,46,34)(38,45,40,43);(10,14)(11,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,40)(27,41)(31,38)(32,34)(33,46)(35,36)(37,47)(39,44)(42,45)(43,48);(10,14)(13,16)(17,20,21,33)(18,25,26,31)(19,28,29,46)(22,37,35,34)(23,38,24,40)(27,45,44,43)(30,47,36,32)(39,48,41,42);(1,3,4,8)(2,6,5,7)(19,29)(23,24)(28,46)(30,36)(32,47)(38,40)(39,41)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7434125);