#476260

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1315 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)(13,14,18,20,19,24,17,23)(15,25,28,38,29,35,27,21)(16,30,33,40,34,36,32,22)(26,37,41,44,42,43,31,39);(5,7)(6,9)(8,12)(10,11)(14,20)(17,18)(21,35)(22,36)(23,24)(25,38)(27,28)(30,40)(31,41)(32,33)(37,44)(39,43);(5,7)(6,10)(8,12)(9,11)(13,15,19,29)(14,21,24,38)(16,31,34,41)(17,28,18,27)(20,35,23,25)(22,39,40,44)(26,33,42,32)(30,43,36,37);(2,4)(7,12)(9,10)(13,16)(14,22)(15,26)(17,33)(18,32)(19,34)(20,36)(21,37)(23,30)(24,40)(25,39)(27,41)(28,31)(29,42)(35,44)(38,43)

Polycyclic presentation:

2^9|a^2=f,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, 383456);