#233049

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1298 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)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(22,27)(23,30)(24,25)(29,34)(31,35)(32,33);(2,4)(21,22,26,29)(23,31,33,35)(24,27,25,34)(28,30,36,32);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,23,26,33)(22,28,29,36)(24,32,25,30)(27,35,34,31);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24,26,25)(22,27,29,34)(23,32,33,30)(28,35,36,31)

Polycyclic presentation:

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

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, 140245);