#233199

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
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

1258 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)(7,13)(8,15)(10,17)(12,19)(14,16)(18,20)(21,22)(23,30)(24,27)(25,28)(26,33)(29,34)(31,36)(32,35);(2,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,18)(15,19)(16,20)(21,23,25,32)(22,26,28,36)(24,31,34,33)(27,30,29,35);(5,7,9,14)(6,10,11,18)(8,16,15,13)(12,20,19,17)(22,27)(23,31)(24,34)(28,29)(30,35)(32,33);(6,11)(10,18)(12,19)(17,20)(22,28)(23,32)(27,29)(31,33)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,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,d^b=di,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140395);