#434724

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 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)(5,6,10,16,11,7,12,15)(8,13,23,29,22,17,21,19)(9,24,25,18,26,30,27,14)(20,32,34,31,35,36,33,28);(5,7)(6,11)(8,19)(9,24)(10,15)(12,16)(13,23)(14,27)(17,21)(18,25)(20,31)(22,29)(26,30)(28,35)(32,33)(34,36);(5,8,12,23,11,22,10,21)(6,13,15,29,7,17,16,19)(9,20,25,33,26,35,27,34)(14,28,24,36,18,31,30,32);(2,4)(5,9)(6,14)(7,18)(8,20)(10,27)(11,26)(12,25)(13,28)(15,24)(16,30)(17,31)(19,32)(21,34)(22,35)(23,33)(29,36)

Polycyclic presentation:

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

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, 341920);