#444534

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351730);