#233365

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#127
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1390 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,27,31)(23,32,24,35)(25,29,26,30)(28,33,36,34);(2,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,19)(14,20)(21,23)(22,28)(24,27)(25,33)(26,34)(29,32)(30,35)(31,36);(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,24,27,23)(22,28,31,36)(25,34,26,33)(29,32,30,35);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,15,20,19)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,32,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140561);