#238937

order	512
nilpotency class	3
isoclinism	isoclinic class of #238935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

870 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=dh,f^c=fhi,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, 146133);