#476262

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1267 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)(8,11)(9,12)(13,14,18,24,20,26,19,25)(15,27,29,22,16,31,30,21)(17,33,35,23,36,41,37,42)(28,39,43,40,32,38,34,44);(5,7)(6,9)(8,12)(10,11)(13,15,20,16)(14,21,26,22)(17,32,36,28)(18,30,19,29)(23,39,42,38)(24,31,25,27)(33,40,41,44)(34,35,43,37);(5,7)(6,10)(8,12)(9,11)(13,16,20,15)(14,22,26,21)(17,34,36,43)(18,29,19,30)(23,40,42,44)(24,27,25,31)(28,35,32,37)(33,38,41,39);(2,4)(7,12)(9,10)(13,17)(14,23)(15,28)(16,32)(18,37)(19,35)(20,36)(21,38)(22,39)(24,33)(25,41)(26,42)(27,40)(29,34)(30,43)(31,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 383458);