#350117

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ei,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, 257313);