#6683507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2311 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)(6,9)(7,11)(13,14,19,24)(15,20,28,36)(16,21,18,23)(17,22,31,39)(25,33,26,34)(27,35,42,44)(29,37,32,40)(30,38,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,36)(16,26,18,25)(17,30,31,41)(21,34,23,33)(22,38,39,43)(27,32,42,29)(35,40,44,37);(1,2,3,4)(15,25)(16,18)(20,33)(21,23)(26,28)(27,30)(29,32)(34,36)(35,38)(37,40)(41,42)(43,44);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,24,23)(15,26,28,25)(17,29,31,32)(20,34,36,33)(22,37,39,40)(27,41,42,30)(35,43,44,38);(6,9)(10,12)(13,17,19,31)(14,22,24,39)(15,27,28,42)(16,29,18,32)(20,35,36,44)(21,37,23,40)(25,30,26,41)(33,38,34,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=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, 6590703);