#193208

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

972 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)(8,9)(12,19)(13,16)(14,17)(18,20)(21,22)(23,31)(24,27)(25,30)(26,29)(28,33)(32,36)(34,35);(1,3,4,2)(5,6)(7,12)(8,11)(9,16)(10,13)(14,18)(15,20)(17,19)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(5,7,10,15)(6,12,13,20)(8,14,9,17)(11,18,16,19)(21,23,26,34)(22,28,29,36)(24,33,25,32)(27,35,30,31);(5,8,10,9)(6,11,13,16)(7,14,15,17)(12,18,20,19)(22,29)(27,30)(28,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100404);