#6673016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,26,25)(16,21,29,32)(17,22,30,31)(23,33,37,27)(24,34,38,39)(28,35,41,42)(36,43,44,40);(6,9)(7,11)(13,15,19,26)(14,20,18,25)(16,27,29,33)(17,24,30,38)(21,23,32,37)(22,34,31,39)(28,40,41,43)(35,36,42,44);(2,4)(13,16,19,29)(14,21,18,32)(15,23,26,37)(17,28,30,41)(20,33,25,27)(22,35,31,42)(24,36,38,44)(34,43,39,40);(6,9)(10,12)(13,17)(14,22)(15,24)(16,28)(18,31)(19,30)(20,34)(21,35)(23,36)(25,39)(26,38)(27,40)(29,41)(32,42)(33,43)(37,44);(5,7,8,11)(6,10,9,12)(14,18)(15,25)(17,30)(20,26)(21,32)(23,27)(24,34)(28,41)(33,37)(36,43)(38,39)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6580212);