#413267

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1096 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320463);