#476209

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

959 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:

(5,6,8,11)(7,10,12,9)(13,14)(15,21)(16,31)(17,23)(18,25)(19,24)(20,26)(22,33)(27,42)(28,39)(29,38)(30,40)(32,44)(34,43)(35,37)(36,41);(1,2)(3,4)(5,7,8,12)(6,9,11,10)(13,15)(14,21)(16,32)(17,34)(18,29)(19,28)(20,30)(22,41)(23,43)(24,39)(25,38)(26,40)(27,35)(31,44)(33,36)(37,42);(2,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,22)(15,17)(18,34)(19,33)(20,27)(21,37)(23,39)(24,42)(25,31)(26,43)(28,32)(29,35)(30,36)(38,41)(40,44);(7,12)(9,10)(13,17,20,36)(14,23,26,41)(15,27,30,16)(18,32,19,35)(21,31,40,42)(22,39,43,38)(24,37,25,44)(28,33,29,34)

Polycyclic presentation:

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

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, 383405);