#322356

order	512
nilpotency class	3
isoclinism	isoclinic class of #322325
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#550
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=fhi,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,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229552);