#6579155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1953 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)(8,10)(9,12)(13,14)(15,20)(16,28)(17,22)(18,31)(19,23)(21,29)(24,38)(25,34)(26,35)(27,41)(30,36)(32,37)(33,40)(39,44)(42,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,25)(14,20,23,34)(16,24,30,39)(17,26,18,27)(21,33,37,43)(22,35,31,41)(28,38,36,44)(29,40,32,42);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,24)(17,29)(18,32)(19,30)(20,33)(22,36)(23,37)(25,39)(26,40)(27,42)(28,31)(34,43)(35,44)(38,41);(1,2)(3,4)(5,7,9,8)(6,11,12,10)(15,25)(20,34)(24,39)(26,27)(33,43)(35,41)(38,44)(40,42);(2,4)(5,8,9,7)(6,10,12,11)(13,17,19,18)(14,22,23,31)(15,26,25,27)(16,29,30,32)(20,35,34,41)(21,36,37,28)(24,40,39,42)(33,44,43,38)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486351);