#273146

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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1046 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)(7,15,18,11)(8,12)(9,16,20,13)(10,14)(17,19)(21,22,25,28)(23,29,30,36)(24,32,34,27)(26,33,35,31);(1,2)(3,4)(5,7,10,18)(6,11,14,15)(8,16,19,13)(9,12,20,17)(21,23,25,30)(22,26,28,35)(24,33,34,31)(27,36,32,29);(2,4)(5,8,10,19)(6,12,14,17)(7,16,18,13)(9,11,20,15);(5,9,10,20)(6,13,14,16)(7,17,18,12)(8,11,19,15)(22,27)(23,30)(24,34)(26,29)(28,32)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,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, 180342);