#6554729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

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, 6461925);