#6552799

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

1803 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)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,33)(26,35)(27,36)(28,37)(31,38)(32,40)(34,39)(41,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,27,19,28,18,29)(14,20,23,37,24,36,22,25)(16,31,33,42,34,26,32,43)(21,38,30,41,40,35,39,44);(2,4)(6,9)(10,12)(13,16,17,33,19,34,18,32)(14,21,23,30,24,40,22,39)(15,26,27,43,28,31,29,42)(20,35,37,44,36,38,25,41);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,32,34,33)(20,25,36,37)(21,30,40,39)(26,42,31,43)(35,44,38,41);(7,11)(10,12)(15,28)(20,36)(25,37)(26,31)(27,29)(35,38)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,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, 6459995);