#134197

order	512
nilpotency class	4
isoclinism	isoclinic class of #133904
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2316 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 688 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)(12,23,25,38)(13,26,27,17)(14,18)(15,21)(16,24)(20,33,34,37)(22,29,36,32)(28,40)(30,39)(31,35);(2,5)(3,7)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,28,30,31)(17,29,26,32)(23,37,38,33)(24,35)(25,34)(27,36)(39,40);(9,12)(10,16,18,30)(11,20)(13,24)(14,25)(15,28,19,31)(17,23,26,38)(21,34)(22,35)(27,39)(29,37,32,33)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,f^b=fg,f^c=fhi,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, 41393);