#444354

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1158 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:

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

Polycyclic presentation:

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

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, 351550);