#131073

order	512
nilpotency class	4
isoclinism	isoclinic class of #130730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38269);