#193082

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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)(5,6,11,14)(7,17,8,19)(9,16,10,15)(12,20,13,18)(22,27)(24,25)(28,35)(29,30)(31,32)(34,36);(1,3,4,2)(5,7,11,8)(6,12,14,13)(9,20,10,18)(15,19,16,17)(21,22)(23,28)(24,27)(25,29)(26,30)(31,35)(32,34)(33,36);(5,8,11,7)(6,13,14,12)(9,20,10,18)(15,19,16,17)(21,23,26,33)(22,28,30,36)(24,32,25,31)(27,34,29,35);(6,14)(7,8)(15,16)(18,20)(21,24,26,25)(22,29,30,27)(23,31,33,32)(28,34,36,35)

Polycyclic presentation:

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

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, 100278);