#193031

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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

1014 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)(7,15)(8,9)(13,14)(16,17)(18,19)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(1,3,4,2)(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,33)(25,31)(27,35)(28,36)(29,32)(30,34);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20)(24,34)(28,32)(29,36)(30,33);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,19,20,18)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100227);