#131072

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	#61406	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 598 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,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, 38268);