#291223

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(9,10,15,18)(11,16,21,22)(12,20,23,24)(13,17,14,19)(25,26,30,34)(27,31,37,40)(28,32,29,33)(35,38,36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,20)(14,24)(15,21)(17,23)(18,22)(25,27,30,37)(26,31,34,40)(28,36,29,35)(32,39,33,38);(9,12,13,16,15,23,14,22)(10,11,19,24,18,21,17,20)(27,35)(28,29)(31,38)(32,33)(36,37)(39,40);(2,6)(5,7)(9,13,15,14)(10,17,18,19)(11,20,21,24)(12,16,23,22)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,40,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=fh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198419);