#425934

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333130);