#134203

order	512
nilpotency class	4
isoclinism	isoclinic class of #133916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2315 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41399);