#272782

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	#37744	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179978);