#135558

order	512
nilpotency class	4
isoclinism	isoclinic class of #135558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1820
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 335 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) = 64

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

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

Polycyclic presentation:

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

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, 42754);