#304978

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

698 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)(11,14)(12,20)(13,18)(15,17)(16,22)(19,24)(21,23)(25,26)(27,33)(28,37)(29,32)(30,38)(31,35)(34,39)(36,40);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,21,23,20)(13,18,15,17)(16,24,19,22)(25,27,29,35)(26,31,32,33)(28,34,38,40)(30,39,37,36);(1,3,5,4)(2,7,8,6)(9,12,13,24,14,23,15,22)(10,16,17,21,11,19,18,20)(25,28,37,26,29,38,30,32)(27,34,39,33,35,40,36,31);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,17,11,18)(12,22,23,24)(16,20,19,21)(26,32)(28,38)(31,33)(34,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,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, 212174);