#129932

order	512
nilpotency class	4
isoclinism	isoclinic class of #129767
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#60688	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

655 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 100 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)(9,10,13,18,15,20,14,19)(11,21,24,34,23,37,25,38)(12,26,29,17,28,36,30,35)(16,27,32,22,31,40,33,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,34)(18,33)(19,32)(20,31)(21,36)(22,30)(26,37)(28,40)(29,39)(35,38);(1,3,5,4)(2,6,7,8)(12,28)(17,35)(21,37)(22,39)(26,36)(27,40)(29,30)(34,38)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,h^b=hi,d^c=di

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, 37128);