#6574507

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1911 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481703);