#316168

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
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

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

normal subgroups only

This group has 462 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)(3,9,11,16)(5,13,10,8)(7,15,14,12)(17,18,24,28,23,29,22,40)(19,30,41,34,33,46,32,25)(20,31,35,44,36,26,42,47)(21,37,43,45,39,48,38,27);(2,6)(7,14)(8,13)(9,16)(17,19,24,41,23,33,22,32)(18,25,40,46,29,34,28,30)(20,27,42,48,36,45,35,37)(21,31,43,44,39,26,38,47);(1,3,10,15,4,11,5,12)(2,7,13,16,6,14,8,9)(17,20,22,35,23,36,24,42)(18,26,40,47,29,31,28,44)(19,27,41,48,33,45,32,37)(21,34,38,46,39,25,43,30);(2,6)(7,14)(8,13)(9,16)(17,21,22,38,23,39,24,43)(18,27,28,37,29,45,40,48)(19,31,32,47,33,26,41,44)(20,34,35,46,36,25,42,30)

Polycyclic presentation:

2^9|a^2=eg,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,g^c=gh

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, 223364);