#500100

order	512
nilpotency class	4
isoclinism	isoclinic class of #499919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 160 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)(11,16)(12,18)(13,26)(14,31)(15,20)(17,22)(19,29)(21,32)(23,34)(24,35)(25,39)(27,37)(28,33)(30,36)(38,40);(2,5)(6,8)(9,11)(10,16)(12,24)(13,27)(14,21)(15,23)(17,36)(18,35)(19,38)(20,34)(22,30)(25,28)(26,37)(29,40)(31,32)(33,39);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,21)(13,28)(14,24)(15,23)(16,26)(18,33)(19,35)(20,37)(22,29)(25,27)(30,40)(31,38)(32,39)(34,36);(2,5)(6,8)(9,12,15,14)(10,18,20,31)(11,21,23,24)(13,25,30,29)(16,32,34,35)(17,33,37,38)(19,26,39,36)(22,28,27,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307037, #72307041, #72307051
maximal normal subgroup of	#72307037, #72307041, #72307051

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407296);