#491561

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33771	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

615 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398757);