#496324

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1104 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403520);