#316166

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=ef,d^2=ei,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, 223362);