#186639

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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×#3475	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93835);