#119408

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6,9,14,10,13,8,12)(7,15,17,19,18,11,16,20)(21,22)(23,33)(24,29,40,48)(25,32)(26,30)(27,31)(28,35)(34,45,51,49)(36,43)(37,44)(38,42,50,52)(39,47,41,46);(1,2)(3,4)(5,7,10,18)(6,11,13,15)(8,17,9,16)(12,20,14,19)(21,23,27,36)(22,28,31,44)(24,38,40,50)(25,37,26,35)(29,45,48,49)(30,43,32,33)(34,39,51,41)(42,46,52,47);(2,4)(6,12,13,14)(11,19,15,20)(21,24,25,39,27,40,26,41)(22,29,30,46,31,48,32,47)(23,34,35,50,36,51,37,38)(28,42,33,49,44,52,43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26604);