#6444774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,35)(25,41)(27,33)(28,44)(29,37)(30,38)(31,39)(32,40)(34,42)(36,43);(6,9)(10,12)(13,15,17,26)(14,19,21,23)(16,28,29,43)(18,24,31,27)(20,36,37,44)(22,33,39,35)(25,32,42,30)(34,40,41,38);(1,2,3,4)(15,24)(18,31)(19,33)(22,39)(23,35)(25,28)(26,27)(30,32)(34,36)(38,40)(41,44)(42,43);(13,16,17,29)(14,20,21,37)(15,25,26,42)(18,32,31,30)(19,34,23,41)(22,40,39,38)(24,28,27,43)(33,36,35,44);(5,7,8,11)(6,10,9,12)(14,21)(19,23)(20,37)(22,39)(33,35)(34,41)(36,44)(38,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351970);