#317959

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#555
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

497 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225155);