#271764

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

957 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)(7,15)(8,17)(9,19)(10,12)(11,16)(13,18)(14,20)(21,22,27,33,25,24,26,29)(23,30,34,35,31,28,32,36);(1,2)(3,4)(5,7,10,8)(6,11,12,13)(9,18,20,16)(14,17,19,15)(21,23)(22,28)(24,30)(25,31)(26,32)(27,34)(29,35)(33,36);(2,4)(6,12)(11,13)(14,19)(15,17)(21,24,25,22)(23,28,31,30)(26,29,27,33)(32,36,34,35);(5,8,10,7)(6,13,12,11)(9,16,20,18)(14,15,19,17)(23,31)(28,30)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=i,d^2=h,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, 178960);