#251226

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1094 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158422);