#193023

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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)(8,17)(9,13)(10,20)(11,14)(12,16)(18,19)(22,26)(24,31)(27,35)(28,32)(29,33)(34,36);(1,3,4,2)(5,7)(6,12)(8,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,22)(23,27)(24,32)(25,28)(26,31)(29,34)(30,36)(33,35);(5,8,11,12)(6,7,14,18)(9,15,10,19)(13,16,20,17)(21,23,25,30)(22,27,28,36)(24,33,31,29)(26,34,32,35);(5,9,11,10)(6,13,14,20)(7,16,18,17)(8,15,12,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100219);