#394673

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2153 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)(3,4)(6,7)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,28)(22,32)(23,37)(25,31)(27,34)(29,33)(30,39)(35,36)(38,40);(5,6)(7,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,22)(17,32)(18,30)(19,21)(20,31)(26,40)(27,38)(28,36)(29,39)(33,35)(34,37);(2,4)(5,7,8,6)(10,17)(11,22)(14,15)(18,26)(19,20)(23,36)(24,25)(28,29)(31,32)(33,34)(37,40)(38,39);(9,12,13,27)(10,18,19,33)(11,23,24,38)(14,29,15,28)(16,30,21,35)(17,26,20,34)(22,36,25,39)(31,37,32,40)

Polycyclic presentation:

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

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, 301869);