#193305

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19)(21,22)(23,31)(24,27)(25,29)(26,30)(28,32)(33,36)(34,35);(1,3,4,2)(5,7)(6,10)(8,17)(9,15)(11,19)(12,20)(13,16)(14,18)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(8,18)(12,16)(13,20)(14,17)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(6,11)(10,19)(12,16)(13,20)(21,24,26,25)(22,29,30,27)(23,32,34,33)(28,31,36,35)

Polycyclic presentation:

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

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, 100501);