#6585522

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(8,12)(9,10)(13,14,18,22)(15,19,26,35)(16,20,30,39)(17,21,31,40)(23,32,25,34)(24,33,27,36)(28,37,29,38)(41,43,42,44);(6,10)(11,12)(13,15,18,26)(14,19,22,35)(16,27,30,24)(17,23,31,25)(20,36,39,33)(21,32,40,34)(28,41,29,42)(37,43,38,44);(1,2,3,4)(5,7,9,8)(6,11,10,12)(15,23)(16,28)(17,31)(19,32)(20,37)(21,40)(24,27)(25,26)(29,30)(33,36)(34,35)(38,39);(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,20,22,39)(15,24,26,27)(17,29,31,28)(19,33,35,36)(21,38,40,37)(23,41,25,42)(32,43,34,44);(5,8,9,7)(6,12,10,11)(15,23)(16,29)(17,31)(19,32)(20,38)(21,40)(25,26)(28,30)(34,35)(37,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^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, 6492718);