#297393

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ei,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,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204589);