#471058

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 378254);