#452793

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

843 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)(9,10,13,18)(11,20)(12,16)(14,17,21,19)(15,23)(22,37,38,30)(24,31)(25,32)(26,40,39,33)(27,35)(28,36)(29,34);(1,3,5,8)(2,6,4,7)(9,11,21,25,13,23,14,24)(10,15,19,31,18,20,17,32)(12,26,29,22,27,39,28,38)(16,33,34,30,35,40,36,37);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,22)(13,27)(14,29)(15,30)(17,34)(18,35)(19,36)(20,37)(21,28)(23,38)(24,39)(25,26)(31,33)(32,40);(1,4,5,2)(3,6,8,7)(10,17,18,19)(15,31,20,32)(16,34,35,36)(30,33,37,40)

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=efi,d^2=e,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, 359989);