#396322

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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)(6,7)(9,10,14,19,16,21,15,20)(11,17,23,33,25,13,24,32)(12,26,29,37,31,36,30,18)(22,38,39,28,40,35,27,34);(5,6,8,7)(9,11)(10,17)(12,27)(13,21)(14,23)(15,24)(16,25)(18,35)(19,33)(20,32)(22,29)(26,34)(28,36)(30,40)(31,39)(37,38);(2,4)(9,12,16,31)(10,18,21,37)(11,22,25,40)(13,28,17,34)(14,30,15,29)(19,36,20,26)(23,27,24,39)(32,38,33,35);(5,7,8,6)(9,13,16,17)(10,11,21,25)(12,28,31,34)(14,33,15,32)(18,22,37,40)(19,24,20,23)(26,27,36,39)(29,38,30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303518);