#510004

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

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

normal subgroups only

This group has 552 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)(9,10,13,18,15,19,30,14)(11,16,22,32,24,33,23,20)(12,25,26,17,29,27,28,34)(21,36,37,31,38,35,39,40);(1,3,5,7)(2,6,8,4)(9,11)(10,16)(12,21)(13,22)(14,20)(15,24)(17,31)(18,32)(19,33)(23,30)(25,36)(26,37)(27,35)(28,39)(29,38)(34,40);(1,4,5,6)(2,7,8,3)(10,14)(11,20)(13,30)(16,23)(17,27)(18,19)(21,35)(22,33)(24,32)(25,34)(26,28)(31,37)(36,38)(39,40);(2,8)(3,7)(9,12)(10,17)(11,21)(13,28)(14,27)(15,29)(16,31)(18,25)(19,34)(20,35)(22,39)(23,37)(24,38)(26,30)(32,36)(33,40)

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=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417200);