#273061

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	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1051 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)(7,13,15,10)(8,16,18,11)(9,12)(14,19)(17,20)(21,22)(23,28)(24,32)(25,27)(26,30)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,10)(8,17)(9,15)(11,20)(12,13)(14,18)(16,19)(22,27)(23,25)(24,33)(28,30)(29,32)(35,36);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,23,26,25)(22,28,30,27)(24,31,33,34)(29,32,36,35);(5,8)(6,11)(7,14)(9,18)(10,19)(12,16)(13,20)(15,17)(21,24,26,33)(22,29,30,36)(23,31,25,34)(27,35,28,32)

Polycyclic presentation:

2^9|a^2=f,c^2=i,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, 180257);