#426777

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(6,9)(10,12)(13,14,18,24,20,26,19,25)(15,27,28,41,30,40,29,21)(16,31,33,39,35,43,34,22)(17,36,37,44,38,23,32,42);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,20,30)(14,21,26,41)(16,32,35,37)(17,33,38,34)(18,29,19,28)(22,42,39,44)(23,43,36,31)(24,40,25,27);(2,4)(5,7,8,11)(6,10,9,12)(13,16,20,35)(14,22,26,39)(15,17,30,38)(18,34,19,33)(21,36,41,23)(24,43,25,31)(27,42,40,44)(28,32,29,37);(6,9)(10,12)(13,17,20,38)(14,23,26,36)(15,16,30,35)(18,32,19,37)(21,39,41,22)(24,44,25,42)(27,31,40,43)(28,34,29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333973);