#271719

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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1050 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)(7,11)(8,10)(9,13)(12,16)(14,17)(15,19)(18,20)(21,22,27,33,25,24,26,29)(23,30,34,35,31,28,32,36);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)(21,23)(22,28)(24,30)(25,31)(26,32)(27,34)(29,35)(33,36);(2,4)(5,7,9,15)(6,11,13,19)(8,14,16,20)(10,17,12,18)(21,24,25,22)(23,28,31,30)(26,29,27,33)(32,36,34,35);(6,10)(8,16)(11,17)(12,13)(14,20)(18,19)(23,31)(28,30)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,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, 178915);