#465641

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1099 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)(18,26)(19,24)(20,25)(21,30)(28,36)(29,37)(31,38)(32,39)(33,41)(34,42)(35,43)(40,44);(6,9)(10,12)(13,15)(14,21,25,38)(16,28)(17,35)(18,30)(19,31)(20,32)(22,40,41,36)(23,43,42,37)(24,27,26,39)(29,33)(34,44);(5,7)(6,10)(8,11)(9,12)(13,16,20,34)(14,22,25,41)(15,28,32,44)(17,18,33,19)(21,36,38,40)(23,24,42,26)(27,43,39,37)(29,31,35,30);(2,4)(7,11)(10,12)(13,17,18,34,20,33,19,16)(14,23,24,41,25,42,26,22)(15,29,30,28,32,35,31,44)(21,37,27,40,38,43,39,36)

Polycyclic presentation:

2^9|b^2=g,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^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372837);