#477202

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1112 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)(3,4)(5,6,9,8)(7,10,11,12)(13,14,19,21,17,23,18,24)(15,25,29,35,26,20,28,37)(16,30,34,40,32,38,33,22)(27,43,31,36,42,39,44,41);(6,8)(10,12)(13,15,18,28,17,26,19,29)(14,20,24,35,23,25,21,37)(16,31,33,27,32,44,34,42)(22,39,38,36,40,43,30,41);(5,7,9,11)(6,10,8,12)(14,21)(15,26)(16,32)(18,19)(20,35)(22,30)(23,24)(25,37)(31,44)(36,43)(38,40)(39,41);(2,4)(5,8,9,6)(7,10,11,12)(13,16)(14,22)(15,27)(17,32)(18,34)(19,33)(20,36)(21,38)(23,40)(24,30)(25,41)(26,42)(28,31)(29,44)(35,39)(37,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384398);