#464639

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1536 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)(13,14,18,24,17,23,19,25)(15,26,29,37,28,20,30,21)(16,31,34,22,33,41,35,40)(27,39,43,42,32,38,44,36);(6,9)(10,12)(13,15,19,30,17,28,18,29)(14,20,25,37,23,26,24,21)(16,27,34,43,33,32,35,44)(22,36,41,39,40,42,31,38);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,28)(14,21,23,37)(16,32,33,27)(18,30,19,29)(20,25,26,24)(22,39,40,38)(31,42,41,36)(34,43,35,44);(7,11)(10,12)(13,16)(14,22)(15,27)(17,33)(18,35)(19,34)(20,36)(21,38)(23,40)(24,31)(25,41)(26,42)(28,32)(29,44)(30,43)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,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, 371835);