#202733

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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)(6,10)(7,14)(8,11)(12,20)(13,18)(15,17)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(1,3,4,2)(5,6,11,18,9,13,8,10)(7,15,19,17,14,12,16,20);(5,7,8,16,9,14,11,19)(6,12,10,17,13,15,18,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109929);