#406899

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,24,29)(19,30,34,45)(20,35,36,48)(21,37,39,47)(22,40,23,28)(25,41,46,33)(26,42,44,32)(27,38,43,31);(2,5)(6,8)(17,19,22,33,24,34,23,41)(18,25,28,30,29,46,40,45)(20,21,32,31,36,39,42,38)(26,47,35,27,44,37,48,43);(9,11,12,16)(10,13,14,15)(17,20,22,32,24,36,23,42)(18,26,28,35,29,44,40,48)(19,31,33,39,34,38,41,21)(25,43,30,47,46,27,45,37);(1,3,4,7)(2,6,5,8)(17,21,22,31,24,39,23,38)(18,27,28,37,29,43,40,47)(19,32,33,36,34,42,41,20)(25,44,30,48,46,26,45,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 314095);