#437425

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344621);