#289416

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=efh,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

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, 196612);