#102501

order	512
nilpotency class	3
isoclinism	isoclinic class of #102469
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	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

454 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9697);