#409253

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

694 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)(11,19,23,31)(12,16)(13,17)(14,18)(15,22,20,26)(21,33,36,39)(24,29)(25,30)(27,32)(28,35,34,37)(38,40);(1,3,4,7)(2,5,6,8)(9,11,13,26,14,23,27,22)(10,15,17,31,18,20,32,19)(12,21,24,37,25,36,38,35)(16,28,29,39,30,34,40,33);(9,10)(11,20,23,15)(12,16)(13,17)(14,18)(19,22,31,26)(21,34,36,28)(24,29)(25,30)(27,32)(33,35,39,37)(38,40);(2,6)(5,8)(9,12,13,24,14,25,27,38)(10,16,17,29,18,30,32,40)(11,21,26,37,23,36,22,35)(15,28,31,39,20,34,19,33)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,c^2=f,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=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, 316449);