#474719

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efgh,c^2=eg,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=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, 381915);