#473250

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

825 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,23,31)(12,16,27,36)(13,28,29,17)(15,22,32,21)(20,33,38,39)(24,30,37,40)(25,35,26,34);(2,5)(6,8)(9,11,14,23)(10,15,18,32)(12,24)(13,22,29,21)(16,33)(17,31,28,19)(20,25)(26,38)(27,37)(30,34)(35,40)(36,39);(2,5)(3,7)(9,12)(10,16)(11,20)(13,25)(14,27)(15,30)(17,34)(18,36)(19,33)(21,24)(22,37)(23,38)(26,29)(28,35)(31,39)(32,40);(1,3,4,7)(2,6,5,8)(12,25)(13,29)(16,34)(17,28)(19,31)(20,24)(21,22)(26,27)(30,33)(35,36)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=efgh,d^2=f,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, 380446);