#7562143

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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2171 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)(13,14,18,23)(15,26,28,21)(16,30,29,38)(17,32,34,41)(19,25,20,24)(22,36,40,35)(27,39,43,44)(31,42,33,37);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,21,23,26)(16,19,29,20)(17,33,34,31)(22,39,40,44)(24,38,25,30)(27,35,43,36)(32,42,41,37);(5,7,9,8)(6,11,10,12)(13,16,20,15,18,29,19,28)(14,21,25,38,23,26,24,30)(17,31,36,27,34,33,35,43)(22,37,41,39,40,42,32,44);(2,4)(5,8,9,7)(6,12,10,11)(13,17,20,36,18,34,19,35)(14,22,25,41,23,40,24,32)(15,27,29,33,28,43,16,31)(21,37,38,39,26,42,30,44);(6,10)(11,12)(14,23)(17,34)(21,26)(24,25)(27,43)(30,38)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,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, 7469339);