#6578203

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#33179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 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)(5,6,9,10)(7,11,8,12)(14,19)(20,35)(21,27)(22,33)(23,37)(36,44)(38,41)(39,40);(6,10)(11,12)(13,14,17,19)(15,23,25,37)(16,27,30,21)(18,33,32,22)(20,34,35,26)(24,41,42,38)(28,44,43,36)(29,40,31,39);(2,4)(13,15,17,25)(14,20,19,35)(16,28,30,43)(18,34,32,26)(21,38,27,41)(22,37,33,23)(24,31,42,29)(36,39,44,40);(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,24)(17,30)(18,29)(19,27)(20,36)(22,40)(23,38)(25,42)(26,28)(31,32)(33,39)(34,43)(35,44)(37,41);(5,8,9,7)(6,12,10,11)(14,22)(16,29)(18,32)(19,33)(20,37)(21,27)(23,35)(24,28)(26,34)(30,31)(38,41)(42,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485399);