#304981

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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

696 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efh,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212177);