#430310

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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	#2684
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^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=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi

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, 337506);