#120638

order	512
nilpotency class	3
isoclinism	isoclinic class of #120636
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=gh,e^c=eh,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, 27834);