#326530

order	512
nilpotency class	3
isoclinism	isoclinic class of #326524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

426 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fi,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, 233726);