#7521990

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429186);