#510001

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=gh,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, 417197);