#463242

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370438);