#510057

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
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=hi

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, 417253);