#151479

order	512
nilpotency class	5
isoclinism	isoclinic class of #151276
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#382	C₂³
prime radical	#537	D₄
LCS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ehi,g^a=gh,d^b=df,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,e^d=eh,g^d=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58675);