#6578849

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,26,34)(16,20,29,37)(17,21,30,38)(23,31,25,33)(24,32,40,43)(27,35,28,36)(39,42,41,44);(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,24)(17,23)(18,26)(20,32)(21,31)(22,34)(25,30)(27,41)(28,39)(29,40)(33,38)(35,44)(36,42)(37,43);(2,4)(5,7,9,8)(6,11,12,10)(15,23)(16,27)(17,30)(19,31)(20,35)(21,38)(24,40)(25,26)(28,29)(32,43)(33,34)(36,37);(5,7,9,8)(6,11,12,10)(13,16)(14,20)(15,24)(17,28)(18,29)(19,32)(21,36)(22,37)(23,39)(25,41)(26,40)(27,30)(31,42)(33,44)(34,43)(35,38);(5,8,9,7)(6,10,12,11)(15,23)(16,28)(17,30)(19,31)(20,36)(21,38)(25,26)(27,29)(33,34)(35,37)(39,41)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6486045);