#219662

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19667	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=e,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,g^c=gh

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, 126858);