#131075

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

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

normal subgroups only

This group has 594 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) = 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)(17,18,22,29,24,28,20,36,23,30,39,48,21,40,38,27)(19,31,34,25,33,26,32,45,35,47,37,44,41,46,42,43);(2,6)(4,12)(7,9)(10,13)(17,19)(18,25)(20,37)(21,33)(22,42)(23,35)(24,41)(26,27)(28,43)(29,31)(30,44)(32,39)(34,38)(36,46)(40,45)(47,48);(1,3,5,11)(2,7,6,9)(4,13,12,10)(8,16,14,15)(17,19)(18,26)(20,32)(21,41)(22,34)(23,35)(24,33)(25,27)(28,47)(29,45)(30,46)(31,40)(36,44)(37,39)(38,42)(43,48)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=ghi,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, 38271);