#6576483

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2707 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)(16,21,33,41)(17,22,31,39)(19,24,32,40)(26,43,44,35)(27,38,30,36)(29,42,34,37);(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,23,25)(16,30,33,27)(17,26,31,44)(19,34,32,29)(21,38,41,36)(22,35,39,43)(24,42,40,37);(2,4)(13,16,18,33)(14,21,23,41)(15,26,28,44)(17,19,31,32)(20,35,25,43)(22,24,39,40)(27,34,30,29)(36,42,38,37);(5,8,9,7)(6,10,11,12)(16,31)(17,33)(19,32)(21,39)(22,41)(24,40)(26,27)(29,34)(30,44)(35,36)(37,42)(38,43);(13,17,18,31)(14,22,23,39)(15,27,28,30)(16,32,33,19)(20,36,25,38)(21,40,41,24)(26,29,44,34)(35,37,43,42)

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,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483679);