#432234

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×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

1362 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)(5,6,9,12)(7,10,8,11)(13,14,18,23)(15,24,29,40)(16,30,33,20)(17,21,25,22)(19,27,36,28)(26,37,42,41)(31,35,43,44)(32,38,34,39);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15,25,27,18,29,17,28)(14,19,21,40,23,36,22,24)(16,31,34,26,33,43,32,42)(20,37,39,35,30,41,38,44);(5,7,9,8)(6,11,12,10)(13,16,18,33)(14,20,23,30)(15,26,29,42)(17,34,25,32)(19,35,36,44)(21,38,22,39)(24,41,40,37)(27,31,28,43);(2,4)(5,8,9,7)(6,11,12,10)(14,21)(15,27)(17,25)(19,36)(20,38)(22,23)(26,31)(28,29)(30,39)(32,34)(35,44)(42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,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, 339430);