#511986

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419182);