#193307

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 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)(7,15)(8,11)(9,13)(10,14)(12,16)(17,20)(18,19)(21,22)(23,27)(24,29)(25,34)(26,30)(28,31)(32,36)(33,35);(1,3,4,2)(6,11)(8,9)(12,19)(13,14)(15,20)(16,17)(21,23)(22,27)(24,33)(25,31)(26,32)(28,34)(29,35)(30,36);(5,7,10,18)(6,12,14,20)(8,16,9,17)(11,19,13,15)(21,23)(22,28)(24,31)(25,33)(26,32)(27,34)(29,36)(30,35);(5,8,10,9)(6,13,14,11)(7,16,18,17)(12,15,20,19)(21,24,26,25)(22,29,30,34)(23,31,32,33)(27,28,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,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, 100503);