#429671

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,17,23,19,24,18,20)(15,25,27,37,29,36,28,21)(16,30,31,22,32,34,33,38)(26,40,41,39,42,35,43,44);(5,7,9,12)(6,10,8,11)(14,20)(16,31)(17,18)(21,25)(22,38)(23,24)(26,41)(27,28)(32,33)(36,37)(39,44)(42,43);(13,15,19,29)(14,21,24,37)(16,26,32,42)(17,28,18,27)(20,25,23,36)(22,35,38,40)(30,44,34,39)(31,43,33,41);(2,4)(5,8,9,6)(7,10,12,11)(13,16,19,32)(14,22,24,38)(15,26,29,42)(17,33,18,31)(20,34,23,30)(21,35,37,40)(25,39,36,44)(27,43,28,41)

Polycyclic presentation:

2^9|a^2=e,b^2=efi,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 336867);