#100745

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

361 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gh,e^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7941);