#213773

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1100 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)(11,18,13,16)(12,23,26,36)(14,20)(15,19)(17,25,33,28)(21,37)(22,34,27,31)(24,40)(29,35)(30,38)(32,39);(9,11,15,13)(10,16,19,18)(12,21,26,24)(14,27,29,22)(17,32,33,30)(20,34,35,31)(23,40,36,37)(25,39,28,38);(9,12)(10,17)(11,21)(13,24)(14,28)(15,26)(16,30)(18,32)(19,33)(20,36)(22,39)(23,35)(25,29)(27,38)(31,37)(34,40);(9,13,15,11)(10,18,19,16)(12,24,26,21)(14,27,29,22)(17,32,33,30)(20,34,35,31)(23,37,36,40)(25,39,28,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=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, 120969);