#453183

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
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

737 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=g,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,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, 360379);