#420733

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327929);