#445428

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

752 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=g,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 352624);