#457561

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

872 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,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, 364757);