#443701

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

900 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,21)(17,23)(18,33)(19,24)(22,26)(25,34)(27,36)(28,37)(29,38)(30,42)(31,39)(32,35)(40,44)(41,43);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,28)(17,32)(18,26)(19,27)(21,37)(22,33)(23,35)(24,36)(25,29)(30,40)(31,41)(34,38)(39,43)(42,44);(6,10)(9,12)(13,16,17,29,19,31,18,30)(14,21,23,38,24,39,33,42)(15,25,26,28,27,40,32,41)(20,34,22,37,36,44,35,43);(1,3,4,2)(13,15)(14,22)(16,25)(17,26)(18,32)(19,27)(20,33)(21,37)(23,36)(24,35)(28,29)(30,41)(31,40)(34,42)(38,44)(39,43)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,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, 350897);