#209154

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#562	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

919 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116350);