#6583375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490571);