#120690

order	512
nilpotency class	3
isoclinism	isoclinic class of #120689
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

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

normal subgroups only

This group has 654 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)(3,4,7,8)(5,6)(9,10)(11,20)(12,25,29,38)(13,30)(14,18)(15,19)(16,22)(17,28,35,27)(21,31,37,40)(23,33)(24,32)(26,34,36,39);(1,3)(2,4,6,8)(5,7)(9,11,13,23,15,24,14,22)(10,16,30,20,19,33,18,32)(12,26,27,37,29,36,28,21)(17,34,38,40,35,39,25,31);(1,4)(2,3,6,7)(5,8)(9,12,13,27,15,29,14,28)(10,17,18,25,19,35,30,38)(11,21,22,36,24,37,23,26)(16,31,20,34,33,40,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gh

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, 27886);