#273056

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	#37769	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1243 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)(23,28)(24,32)(25,27)(26,30)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,10)(12,18)(13,14)(16,17)(19,20)(22,27)(23,25)(24,33)(28,30)(29,32)(35,36);(2,4)(5,8,10,17)(6,12,14,20)(7,15,9,16)(11,18,13,19)(21,23,26,25)(22,28,30,27)(24,31,33,34)(29,32,36,35);(5,9,10,7)(6,13,14,11)(8,16,17,15)(12,19,20,18)(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=h,d^2=h,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, 180252);