#428648

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1013 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ei,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=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, 335844);