#7565200

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,28)(17,33)(18,23)(19,34)(20,24)(22,31)(25,40)(26,36)(27,37)(29,43)(30,39)(32,44)(35,41)(38,42);(1,2)(3,4)(6,10)(11,12)(13,15,20,27)(14,21,24,37)(16,29,32,41)(17,18,26,19)(22,38,39,40)(23,36,34,33)(25,31,42,30)(28,43,44,35);(5,7,9,8)(6,11,10,12)(13,16,18,30,20,32,19,31)(14,22,34,44,24,39,23,28)(15,25,17,29,27,42,26,41)(21,35,36,38,37,43,33,40);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,15,20,26,18,27)(14,21,23,33,24,37,34,36)(16,29,31,25,32,41,30,42)(22,35,28,40,39,43,44,38);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,23,24,34)(15,26,27,17)(16,30,32,31)(21,36,37,33)(22,28,39,44)(25,41,42,29)(35,38,43,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472396);