#187619

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94815);