#422235

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1026 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)(6,12,15,24)(7,16,19,25)(14,20,29,35)(18,30,33,36);(5,6,10,12,11,15,9,24)(7,17,16,26,19,13,25,28)(8,20,22,29,23,35,21,14)(18,32,30,31,33,34,36,27);(5,7)(6,13)(8,18)(9,16)(10,25)(11,19)(12,26)(14,27)(15,17)(20,34)(21,30)(22,36)(23,33)(24,28)(29,31)(32,35);(2,4)(5,8,9,21,11,23,10,22)(6,14,12,20,15,29,24,35)(7,18,16,30,19,33,25,36)(13,27,26,34,17,31,28,32)

Polycyclic presentation:

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

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, 329431);