#101814

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
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	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

386 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=gi,d^2=gh,e^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

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, 9010);