#7614190

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=h,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, 7521386);