#401107

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308303);