#449761

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1446 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)(6,9)(10,12)(13,14,18,23,19,21,17,31)(15,20,27,35,26,36,24,33)(16,28,30,37,29,38,32,22)(25,40,39,43,42,44,41,34);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,24)(18,27)(19,26)(21,36)(22,34)(23,35)(28,40)(29,42)(30,39)(31,33)(32,41)(37,43)(38,44);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,24)(17,18)(20,33)(22,37)(25,39)(26,27)(30,32)(34,40)(35,36)(41,42)(43,44);(6,9)(10,12)(13,16,19,29)(14,22,21,37)(15,25,26,42)(17,30,18,32)(20,34,36,43)(23,38,31,28)(24,39,27,41)(33,40,35,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356957);