#265752

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

856 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:

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

Polycyclic presentation:

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

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, 172948);