#439263

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 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)(11,16)(12,26)(13,30)(14,33)(15,20)(17,28)(18,32)(19,21)(22,38)(23,37)(24,35)(25,36)(27,39)(29,40)(31,34);(1,3)(2,6)(4,7)(5,8)(9,11,21,23,15,25,14,24)(10,16,19,37,20,36,33,35)(12,27,28,31,29,13,22,32)(17,34,40,30,38,18,26,39);(1,2,5,4)(3,6,8,7)(9,12,14,28,15,29,21,22)(10,17,19,26,20,38,33,40)(11,13,23,31,25,27,24,32)(16,34,35,30,36,18,37,39);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,22)(12,23)(14,31)(15,27)(16,26)(17,37)(19,39)(20,34)(21,32)(24,29)(25,28)(30,33)(35,38)(36,40)

Polycyclic presentation:

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

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, 346459);