#213926

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18,21,26,22,25,36,40)(19,27,30,41,29,23,34,38)(20,31,32,42,33,24,35,39)(28,37,45,47,44,43,46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,27)(26,41)(30,36)(31,43)(32,46)(33,44)(35,45)(38,40)(39,48)(42,47);(1,4,5,12)(2,7,8,11)(3,9,10,16)(6,13,14,15)(23,27)(37,43)(38,41)(47,48);(3,10)(6,14)(9,16)(13,15)(17,20,22,33)(18,24,25,31)(19,28,29,44)(21,35,36,32)(23,37,27,43)(26,42,40,39)(30,46,34,45)(38,48,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121122);