#135501

order	512
nilpotency class	4
isoclinism	isoclinic class of #135501
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1820
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18)(19,28,31,43)(20,23)(21,34,35,26)(22,27)(24,30)(25,33)(29,41,45,42)(32,40,47,37)(36,44)(38,48)(39,46);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(18,23,27,25)(24,36,39,38)(26,37,34,40)(28,41,43,42);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(17,19,20,29,22,31,33,45)(18,24,25,38,27,39,23,36)(21,30,32,44,35,46,47,48)(26,28,40,42,34,43,37,41)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=d,d^2=i,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fgi,f^c=fgh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42697);