#6579227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3277	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6486423);