#465639

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1157 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)(15,27)(16,23)(17,22)(18,25)(19,26)(20,24)(21,30)(28,36)(29,37)(31,38)(32,39)(33,43)(34,40)(35,41)(42,44);(6,9)(10,12)(13,15)(14,21,26,39)(16,33)(17,29)(18,30)(19,31)(20,32)(22,36,41,43)(23,37,40,42)(24,27,25,38)(28,34)(35,44);(5,7)(6,10)(8,11)(9,12)(13,16,19,34)(14,22,26,41)(15,28,31,33)(17,18,35,20)(21,36,39,43)(23,24,40,25)(27,42,38,37)(29,32,44,30);(2,4)(7,11)(10,12)(13,17,18,34,19,35,20,16)(14,23,24,41,26,40,25,22)(15,29,30,28,31,44,32,33)(21,37,27,43,39,42,38,36)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=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, 372835);