#273170

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1092 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)(7,13,15,11)(8,10)(9,16,18,12)(14,19)(17,20)(21,22)(23,32)(24,29)(25,31)(26,35)(27,30)(28,34)(33,36);(1,2)(3,4)(5,7,10,15)(6,11,8,13)(9,17,18,14)(12,20,16,19)(21,23,27,24)(22,28,30,36)(25,34,26,33)(29,35,32,31);(2,4)(5,8,10,6)(7,13,15,11)(9,16,18,12)(14,20,17,19)(21,24,27,23)(22,29,30,32)(25,34,26,33)(28,35,36,31);(5,9,10,18)(6,12,8,16)(7,14,15,17)(11,19,13,20)(21,25)(22,30)(23,33)(24,34)(26,27)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180366);