#104424

order	512
nilpotency class	3
isoclinism	isoclinic class of #104373
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	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

418 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11620);