#424379

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=ef,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,g^c=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, 331575);