#328425

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

776 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)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24)(25,26,29,33)(27,30,34,39)(28,31,37,32)(35,38,36,40);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23)(25,27,29,34)(26,30,33,39)(28,36,37,35)(31,40,32,38);(3,7)(6,8)(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22)(26,31)(28,37)(30,38)(32,33)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(27,34)(30,39)(35,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235621);