#508779

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

596 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,6)(4,5)(7,8)(9,10,14,20,16,22,13,18)(11,23,27,17,28,35,26,36)(12,29,30,21,31,19,15,32)(24,33,39,38,40,34,25,37);(2,5)(6,8)(9,11)(10,17)(12,25)(13,27)(14,26)(15,24)(16,28)(18,35)(19,34)(20,23)(21,37)(22,36)(29,33)(30,40)(31,39)(32,38);(3,7)(6,8)(10,18)(11,24)(12,15)(13,14)(17,33)(19,29)(20,22)(23,38)(25,27)(26,39)(28,40)(30,31)(34,36)(35,37);(1,3,4,7)(2,6,5,8)(9,12,13,15,16,31,14,30)(10,19,18,21,22,29,20,32)(11,25,27,24,28,39,26,40)(17,34,35,37,36,33,23,38)

Polycyclic presentation:

2^9|a^2=h,d^2=efhi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi

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, 415975);