#511748

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3124 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 3124 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287145
maximal normal subgroup of	#72287145
maximal quotient of	#72287653, #72287672, #72287727, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418944);