#408081

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

731 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)(9,10,15,20)(11,21)(12,26,29,17)(13,18,31,28)(14,19,30,36)(16,32)(22,23)(24,35)(25,37)(27,40)(33,34)(38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,27)(13,23)(14,32)(15,25)(17,33)(18,34)(19,37)(20,35)(21,36)(22,29)(24,30)(26,39)(28,38)(31,40);(9,12,14,28,15,29,30,18)(10,17,19,31,20,26,36,13)(11,22,24,38,25,27,32,34)(16,33,21,23,35,39,37,40);(2,6)(5,8)(9,13,30,26,15,31,14,17)(10,18,36,29,20,28,19,12)(11,23,24,39,25,40,32,33)(16,34,21,22,35,38,37,27)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,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, 315277);