#350112

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257308);