#102504

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

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

normal subgroups only

This group has 430 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,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, 9700);