#407691

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314887);