#103344

order	512
nilpotency class	3
isoclinism	isoclinic class of #103301
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

522 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,45,48,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28)(33,35,38,46)(34,39,42,48)(36,45,37,44)(40,47,41,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,10,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=ei

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, 10540);