#6682791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2210 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)(13,14,18,19)(15,23,26,20)(16,21,29,33)(17,22,32,31)(24,27,39,34)(25,37,36,30)(28,35,43,42)(38,40,41,44);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,19,23)(16,27,29,34)(17,30,32,37)(21,24,33,39)(22,36,31,25)(28,41,43,38)(35,40,42,44);(1,2,3,4)(13,16)(14,21)(15,24)(17,28)(18,29)(19,33)(20,34)(22,35)(23,27)(25,38)(26,39)(30,44)(31,42)(32,43)(36,41)(37,40);(6,10)(9,11)(13,14,18,19)(15,23,26,20)(16,21,29,33)(17,31,32,22)(24,27,39,34)(25,30,36,37)(28,42,43,35)(38,44,41,40);(7,12)(9,11)(13,17,18,32)(14,22,19,31)(15,25,26,36)(16,28,29,43)(20,30,23,37)(21,35,33,42)(24,38,39,41)(27,40,34,44)

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=dfi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589987);