#497818

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405014);