#234785

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1456 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)(6,10)(7,8)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,31)(24,32)(25,33)(26,35)(27,36)(28,37)(29,38)(30,39)(34,41)(40,43)(42,44);(5,6)(7,10)(8,12)(9,11)(13,15,18,25)(14,19,22,33)(16,23,27,41)(17,28,30,24)(20,34,36,31)(21,37,39,32)(26,42,29,40)(35,43,38,44);(2,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,23)(17,29)(18,27)(19,31)(21,38)(22,36)(24,42)(25,41)(26,30)(28,40)(32,44)(33,34)(35,39)(37,43);(6,11)(10,12)(13,17,18,30)(14,21,22,39)(15,24,25,28)(16,26,27,29)(19,32,33,37)(20,35,36,38)(23,40,41,42)(31,43,34,44)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,d^c=di,e^c=ei

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, 141981);