#6576531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2667 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,25,28,20)(16,30,34,21)(17,35,32,22)(19,24,33,41)(26,37,44,43)(27,38,31,40)(29,42,36,39);(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,23,25)(16,31,34,27)(17,26,32,44)(19,36,33,29)(21,40,30,38)(22,37,35,43)(24,42,41,39);(2,4)(13,16,18,34)(14,21,23,30)(15,26,28,44)(17,19,32,33)(20,37,25,43)(22,24,35,41)(27,36,31,29)(38,42,40,39);(5,8,9,7)(6,10,11,12)(16,32)(17,34)(19,33)(21,35)(22,30)(24,41)(26,27)(29,36)(31,44)(37,38)(39,42)(40,43);(13,17,18,32)(14,22,23,35)(15,27,28,31)(16,33,34,19)(20,38,25,40)(21,41,30,24)(26,29,44,36)(37,39,43,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483727);