#239119

order	512
nilpotency class	3
isoclinism	isoclinic class of #233245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3184	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146315);