#316186

order	512
nilpotency class	3
isoclinism	isoclinic class of #316170
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

451 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=ei,c^2=ef,d^2=ehi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=di,g^c=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, 223382);