#395721

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1875 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)(5,6)(7,8)(9,10,13,18,14,19,29,31)(11,15,21,32,23,33,22,17)(12,24,26,36,27,37,28,16)(20,35,38,40,39,34,25,30);(5,7,8,6)(9,11)(10,15)(12,25)(13,21)(14,23)(16,34)(17,31)(18,32)(19,33)(20,26)(22,29)(24,30)(27,38)(28,39)(35,36)(37,40);(2,4)(6,7)(9,12,14,27)(10,16,19,36)(11,20,23,39)(13,28,29,26)(15,30,33,40)(17,35,32,34)(18,37,31,24)(21,25,22,38);(9,11)(10,17)(12,20)(13,22)(14,23)(15,31)(16,35)(18,33)(19,32)(21,29)(24,30)(25,26)(27,39)(28,38)(34,36)(37,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 302917);