#153076

order	512
nilpotency class	5
isoclinism	isoclinic class of #149931
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×C₄⋊Q₈	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 277 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)(9,10,13,20,15,19,29,38)(11,22,24,25,17,16,33,12)(14,21,31,34,32,18,28,37)(23,35,40,39,30,36,27,26);(2,5)(6,7)(9,11,13,27,15,17,29,40)(10,16,34,25,19,22,37,12)(14,30,31,33,32,23,28,24)(18,36,20,39,21,35,38,26);(1,3,4,8)(2,6,5,7)(9,12,15,25)(10,17,19,11)(13,16,29,22)(14,26,32,39)(18,23,21,30)(20,24,38,33)(27,34,40,37)(28,35,31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=hi,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=dghi,e^b=ei,f^b=fgh,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, 60272);