#271763

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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:

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

Polycyclic presentation:

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

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, 178959);