#186688

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
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×#3522	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2392 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)(5,6,9,13,10,11,8,18)(7,14,17,12,16,20,15,19)(21,22)(23,32)(24,35)(25,31)(26,30)(27,29)(28,34)(33,36);(2,4)(6,11)(8,9)(12,19)(15,17)(21,23)(22,28)(24,29)(25,35)(26,34)(27,33)(30,36)(31,32);(5,7,10,16)(6,12,11,19)(8,15,9,17)(13,20,18,14)(21,24,27,28)(22,23,29,33)(25,32,26,36)(30,35,31,34);(5,8,10,9)(6,13,11,18)(7,15,16,17)(12,20,19,14)(22,29)(23,33)(30,31)(34,35)

Polycyclic presentation:

2^9|a^2=g,c^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,d^b=dh,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, 93884);