#203853

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

646 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)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22)(23,30)(24,25)(26,29)(27,31)(28,34)(32,36)(33,35);(1,3,4,2)(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23)(22,27)(24,30)(25,32)(26,33)(28,35)(29,36)(31,34);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20)(21,24,34,29,26,28,25,22)(23,30,31,36,33,35,32,27);(7,14)(11,18)(15,20)(17,19)(21,25,26,34)(22,24,29,28)(23,31,33,32)(27,35,36,30)

Polycyclic presentation:

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

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, 111049);