#191295

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98491);