#219713

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

570 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,d^b=di,f^b=fh,d^c=di,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, 126909);