#111293

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18489);