#233271

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
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

1292 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)(7,10)(8,11)(9,12)(13,16)(14,19)(15,17)(18,20)(22,27)(23,30)(24,25)(29,34)(31,35)(32,33);(2,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20)(21,22,26,29)(23,31,33,35)(24,34,25,27)(28,30,36,32);(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,23,26,33)(22,28,29,36)(24,30,25,32)(27,35,34,31);(6,12)(7,14)(11,13)(15,20)(22,29)(27,34)(28,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140467);