#413857

order	512
nilpotency class	3
isoclinism	isoclinic class of #411625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups; only the 262 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 262 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321053);