#415686

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 322882);