#272754

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	#37761	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1223 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)(8,12)(10,14)(15,18,16,19)(17,20)(21,22,26,31)(23,32,24,28)(25,29,35,30)(27,34,36,33);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,15,17,16)(12,18,20,19)(21,23,26,24)(22,27,31,36)(25,34,35,33)(28,29,32,30);(2,4)(5,8,10,17)(6,12,14,20)(7,15,9,16)(11,18,13,19)(21,24,26,23)(22,28,31,32)(25,34,35,33)(27,29,36,30);(5,9,10,7)(6,13,14,11)(8,16,17,15)(12,19,20,18)(22,29)(25,35)(27,32)(28,36)(30,31)(33,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^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, 179950);