#409768

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 316964);