#202966

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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)(5,6,9,12)(7,10,14,19)(8,11,16,13)(15,17,20,18)(22,27)(23,31)(24,25)(28,35)(29,30)(32,33);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,17)(12,19)(13,18)(16,20)(21,22,24,29,26,30,25,27)(23,28,33,36,32,35,31,34);(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,23,25,31,26,32,24,33)(22,28,27,34,30,35,29,36);(6,12)(7,14)(11,13)(15,20)(21,24,26,25)(22,29,30,27)(23,31,32,33)(28,34,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dhi,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110162);