#136466

order	512
nilpotency class	4
isoclinism	isoclinic class of #136369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ehi

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, 43662);