#214026

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121222);