#119024

order	512
nilpotency class	3
isoclinism	isoclinic class of #119021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

562 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26220);