#100706

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
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

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

normal subgroups only

This group has 353 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gh,d^2=gh,e^2=gi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7902);