#414584

order	512
nilpotency class	3
isoclinism	isoclinic class of #411625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1009 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 206 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gh,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 321780);