#303751

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

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, 210947);