#453282

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

715 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)(9,10,13,18)(11,15)(12,25)(14,17,20,19)(16,27)(21,37,38,30)(22,32)(23,31)(24,36)(26,40,39,33)(28,34)(29,35);(1,2,4,5)(3,6,7,8)(9,11,20,24,13,22,14,23)(10,15,19,36,18,32,17,31)(12,26,29,21,27,39,28,38)(16,33,34,30,25,40,35,37);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,21)(13,27)(14,29)(15,30)(17,34)(18,25)(19,35)(20,28)(22,38)(23,39)(24,26)(31,40)(32,37)(33,36);(2,5)(6,8)(10,17,18,19)(11,22)(15,31,32,36)(16,34,25,35)(21,38)(23,24)(26,39)(30,40,37,33)

Polycyclic presentation:

2^9|a^2=fgh,b^2=efg,c^2=efi,d^2=f,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, 360478);