#102872

order	512
nilpotency class	3
isoclinism	isoclinic class of #102857
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#392
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

443 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10068);