#500078

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1555 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407274);