#413461

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,27)(13,32)(14,20)(15,21)(16,24)(17,29)(18,23)(19,33)(25,35)(26,34)(28,36)(30,38)(31,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,16,21,35)(12,28,31,13)(14,25,33,24)(17,36,38,32)(18,27,39,37)(19,34,20,22)(23,30,40,29);(1,4,5,3)(2,6,7,8)(9,12,15,31)(10,17,21,38)(11,23,26,40)(13,25,28,24)(14,30,33,29)(16,18,35,39)(19,37,20,27)(22,36,34,32);(9,13,14,23,15,28,33,40)(10,18,19,32,21,39,20,36)(11,12,25,30,26,31,24,29)(16,27,34,17,35,37,22,38)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=h,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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 320657);