#153072

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2326 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60268);