#354237

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261433);