#430311

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
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

1169 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:

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

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=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, 337507);