#273065

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

968 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)(7,15,17,11)(8,12)(9,19,20,13)(10,14)(16,18)(21,22)(23,30)(24,26)(25,29)(27,34)(28,33)(31,35)(32,36);(1,2)(3,4)(5,7)(6,11)(8,13)(9,12)(10,17)(14,15)(16,20)(18,19)(22,26)(23,31)(24,33)(27,30)(28,29)(35,36);(2,4)(5,8,10,18)(6,12,14,16)(7,13,17,19)(9,15,20,11);(5,9)(6,13)(7,16)(8,15)(10,20)(11,18)(12,17)(14,19)(21,23,25,31)(22,27,29,36)(24,34,33,32)(26,35,28,30)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,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, 180261);