#496221

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1712 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1712 conjugacy classes of subgroups — too many to draw. Only its 206 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)(9,10,14,20,15,21,12,18)(11,16,22,30,24,32,25,34)(13,17,26,33,27,19,28,35)(23,29,38,40,37,31,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,29)(18,34)(19,31)(20,30)(21,32)(26,38)(27,37)(28,36)(33,40)(35,39);(3,8)(5,7)(10,17)(12,14)(13,27)(16,29)(18,35)(19,21)(20,33)(22,25)(23,37)(30,40)(31,32)(34,39);(2,6)(5,7)(9,12,15,14)(10,18,21,20)(11,22,24,25)(13,26,27,28)(16,30,32,34)(17,33,19,35)(23,36,37,38)(29,39,31,40)

Polycyclic presentation:

2^9|a^2=g,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^b=di,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, 403417);