#439565

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1119 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=efi,d^2=ei,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, 346761);