#305018

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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)(9,10,13,16)(11,17,18,24)(12,21,19,15)(14,22,23,20)(25,26,29,28)(27,33,35,31)(30,37,38,32)(34,39,40,36);(1,3,5,4)(2,6,8,7)(11,18)(12,19)(15,21)(17,24)(25,27,29,35)(26,31,28,33)(30,39,38,36)(32,40,37,34);(1,3,5,4)(2,7,8,6)(9,11,19,22,13,18,12,20)(10,14,21,24,16,23,15,17)(25,28,29,26)(27,34)(30,37,38,32)(31,39)(33,36)(35,40);(1,4,5,3)(2,7,8,6)(26,28)(31,33)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=efh,d^2=ehi,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212214);