#409654

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 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)(3,5)(4,6)(7,8)(9,10,13,27,14,17,28,18)(11,19,23,30,24,15,22,20)(12,16,25,38,26,31,39,32)(21,33,36,40,37,29,35,34);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,22)(14,24)(16,29)(17,19)(18,20)(23,28)(25,35)(26,37)(27,30)(31,33)(32,34)(36,39)(38,40);(9,10)(11,20)(12,16)(13,27)(14,17)(15,22)(18,28)(19,23)(21,34)(24,30)(25,38)(26,31)(29,35)(32,39)(33,36)(37,40);(2,6)(5,8)(9,12,13,25,14,26,28,39)(10,16,27,38,17,31,18,32)(11,21,22,35,24,37,23,36)(15,29,30,40,19,33,20,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316850);