#237892

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209972
maximal normal subgroup of	#72209972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145088);