#473025

order	512
nilpotency class	3
isoclinism	isoclinic class of #456884
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

841 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380221);