#6685183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 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:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,26,33)(16,20,27,34)(17,22,29,36)(23,30,25,32)(24,31,39,43)(28,35,40,44)(37,41,38,42);(6,9)(7,11)(13,15,18,26)(14,19,21,33)(16,24,27,39)(17,23,29,25)(20,31,34,43)(22,30,36,32)(28,37,40,38)(35,41,44,42);(1,2,3,4)(15,23)(16,27)(17,29)(19,30)(20,34)(22,36)(24,38)(25,26)(31,42)(32,33)(37,39)(41,43);(6,9)(10,12)(13,16,18,27)(14,20,21,34)(15,24,26,39)(17,28,29,40)(19,31,33,43)(22,35,36,44)(23,37,25,38)(30,41,32,42);(5,7,8,11)(6,10,9,12)(14,21)(15,23)(17,29)(19,32)(20,34)(24,37)(25,26)(28,40)(30,33)(31,42)(38,39)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^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,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592379);