#464637

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

1488 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)(7,10)(8,9)(11,12)(13,14,19,15,18,23,20,24)(16,27,30,37,29,25,31,21)(17,26,34,22,33,40,35,39)(28,38,43,42,32,36,44,41);(6,9)(10,12)(13,15,19,23,18,24,20,14)(16,21,30,27,29,37,31,25)(17,26,35,39,33,40,34,22)(28,36,44,42,32,38,43,41);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,29)(14,21,23,37)(15,25,24,27)(17,32,33,28)(19,31,20,30)(22,38,39,36)(26,42,40,41)(34,43,35,44);(7,11)(10,12)(13,17)(14,22)(15,26)(16,28)(18,33)(19,35)(20,34)(21,36)(23,39)(24,40)(25,41)(27,42)(29,32)(30,44)(31,43)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 371833);