#512367

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419563);