#214635

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 904 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=hi,f^2=h,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, 121831);