#129323

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 851 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,7,8,5)(4,6)(9,10,14,21,16,19,33,34,15,22,32,40,13,20,24,39)(11,23,26,38,25,17,29,36,27,35,31,18,30,37,12,28);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,30)(14,26)(15,27)(16,25)(18,34)(19,35)(20,23)(21,36)(22,37)(28,40)(29,33)(31,32)(38,39);(9,12,16,26,15,29,13,31)(10,18,19,28,22,38,20,36)(11,24,25,14,27,33,30,32)(17,34,35,40,37,39,23,21)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36519);