#6584889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492085);