#463240

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,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, 370436);