#100702

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

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

normal subgroups only

This group has 357 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 7898);