#138121

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

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

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

Polycyclic presentation:

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

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, 45317);