#433287

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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

1081 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)(6,10)(11,12)(13,14,18,23,19,25,33,24)(15,22,17,32,28,39,27,20)(16,21,30,36,31,38,43,37)(26,35,29,42,41,44,40,34);(1,3,4,2)(5,6)(7,11)(8,12)(9,10)(13,15)(14,20)(16,29)(17,33)(18,27)(19,28)(21,35)(22,24)(23,39)(25,32)(26,30)(31,40)(34,36)(37,42)(38,44)(41,43);(2,3)(5,7,9,8)(6,11,10,12)(13,16,18,30,19,31,33,43)(14,21,23,36,25,38,24,37)(15,26,17,29,28,41,27,40)(20,34,22,35,32,42,39,44);(5,8,9,7)(6,11,10,12)(13,17,33,15,19,27,18,28)(14,22,24,20,25,39,23,32)(16,29,43,26,31,40,30,41)(21,35,37,34,38,44,36,42)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,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, 340483);