#396930

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1024 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)(11,21)(12,25)(13,29)(14,20)(15,19)(16,23)(17,24)(18,27)(22,33)(26,34)(28,39)(30,32)(31,40)(35,37)(36,38);(1,3,4,7)(2,5,6,8)(9,11,15,30,14,24,13,23)(10,16,29,17,20,32,19,21)(12,22,28,40,27,37,26,36)(18,31,39,33,25,38,34,35);(2,6)(5,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,30)(16,19)(18,33)(20,21)(25,35)(26,36)(27,37)(28,40)(29,32)(31,34)(38,39);(9,12,13,26,14,27,15,28)(10,18,19,34,20,25,29,39)(11,22,23,36,24,37,30,40)(16,31,21,35,32,38,17,33)

Polycyclic presentation:

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

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, 304126);