#291943

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 199139);