#357521

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4406 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)(9,10,14,18,15,19,13,27)(11,20,23,16,22,32,28,31)(12,17,25,35,26,36,29,34)(21,37,39,30,38,33,24,40);(2,4)(9,11)(10,16)(12,24)(13,28)(14,23)(15,22)(17,33)(18,32)(19,31)(20,27)(21,25)(26,39)(29,38)(30,34)(35,40)(36,37);(6,8)(9,12)(10,17)(11,21)(13,29)(14,25)(15,26)(16,30)(18,35)(19,36)(20,37)(22,38)(23,39)(24,28)(27,34)(31,40)(32,33);(11,22)(16,31)(20,32)(21,38)(23,28)(24,39)(30,40)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264717);