#6682337

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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2222 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)(7,11)(8,10)(9,12)(13,14,19,18)(15,20,26,24)(16,21,29,33)(17,30,32,22)(23,34,38,27)(25,31,39,35)(28,41,42,36)(37,40,44,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,18,24)(16,27,29,34)(17,31,32,35)(21,23,33,38)(22,25,30,39)(28,37,42,44)(36,43,41,40);(1,2,3,4)(13,16)(14,21)(15,23)(17,28)(18,33)(19,29)(20,34)(22,36)(24,27)(25,37)(26,38)(30,41)(31,40)(32,42)(35,43)(39,44);(6,10)(9,11)(13,14,19,18)(15,24,26,20)(16,21,29,33)(17,30,32,22)(23,27,38,34)(25,35,39,31)(28,41,42,36)(37,43,44,40);(7,12)(9,11)(13,17,19,32)(14,22,18,30)(15,25,26,39)(16,28,29,42)(20,35,24,31)(21,36,33,41)(23,37,38,44)(27,40,34,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6589533);