#6585691

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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)(5,6,9,10)(7,11,8,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,34)(24,33)(25,35)(26,32)(27,38)(28,40)(29,39)(30,36)(31,37)(41,44)(42,43);(6,10)(11,12)(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)(5,7,9,8)(6,11,10,12)(14,20)(15,23)(16,28)(18,31)(19,32)(21,39)(24,27)(25,26)(29,30)(34,35)(36,40)(43,44);(5,7,9,8)(6,11,10,12)(13,16,17,30)(14,21,20,36)(15,24,25,27)(18,29,31,28)(19,33,35,38)(22,39,37,40)(23,41,26,42)(32,43,34,44);(5,8,9,7)(6,12,10,11)(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=f,b^2=i,c^2=fgi,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, 6492887);