#404673

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

919 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)(3,4)(5,6,10,15)(7,17)(8,13)(9,14,26,22)(11,16,28,31)(12,21)(18,32,34,27)(19,24)(20,33)(23,29)(25,30)(35,36);(5,7)(6,12,15,17)(8,22,24,14)(9,19)(10,21)(11,20)(13,26)(16,23,31,33)(18,25,34,36)(27,35)(28,29)(30,32);(5,8,10,24)(6,13)(7,18)(9,23,26,33)(11,25,28,36)(12,27,17,32)(14,29)(15,19)(16,30)(20,22)(21,34)(31,35);(2,4)(5,9,11,27,10,26,28,32)(6,14,16,18,15,22,31,34)(7,19,20,35,21,13,29,30)(8,23,25,17,24,33,36,12)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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, 311869);