#187631

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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×#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2070 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)(5,6,10,12,9,8,16,17)(7,11,15,19,14,13,18,20)(22,27,29,32)(28,35,36,34);(5,7,9,14)(6,11,8,13)(10,18,16,15)(12,20,17,19)(21,22)(23,28)(24,27)(25,32)(26,29)(30,34)(31,36)(33,35);(2,4)(5,8)(6,9)(7,13)(10,12)(11,14)(15,19)(16,17)(18,20)(21,23)(22,28)(24,30)(25,33)(26,31)(27,34)(29,36)(32,35);(7,14)(11,13)(15,18)(19,20)(21,24,26,25)(22,27,29,32)(23,30,31,33)(28,34,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,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, 94827);