#464054

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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1453 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)(15,21)(16,31)(17,33)(18,24)(19,23)(20,25)(22,35)(26,42)(27,43)(28,40)(29,39)(30,41)(32,37)(34,36)(38,44);(6,9)(10,12)(13,15,18,28,20,30,19,29)(14,21,24,40,25,41,23,39)(16,32,33,43,22,38,34,26)(17,27,35,44,36,42,31,37);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,20,22)(14,17,25,36)(15,26,30,43)(18,34,19,33)(21,37,41,44)(23,35,24,31)(27,40,42,39)(28,38,29,32);(7,11)(10,12)(13,17)(14,22)(15,27)(16,25)(18,35)(19,31)(20,36)(21,38)(23,33)(24,34)(26,40)(28,44)(29,37)(30,42)(32,41)(39,43)

Polycyclic presentation:

2^9|b^2=g,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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, 371250);