#151469

order	512
nilpotency class	5
isoclinism	isoclinic class of #151256
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	#390	C₂³
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #390 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 359 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)(17,18)(19,23)(20,24,32,42)(21,26)(22,25)(27,45,30,48)(28,38)(29,39)(31,43,33,41)(34,37)(35,44)(36,47,40,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,29)(18,23,25,39)(20,30,32,27)(21,34,35,28)(24,40,42,36)(26,37,44,38)(31,46,33,47)(41,45,43,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,21,31,22,32,35,33)(18,24,26,41,25,42,44,43)(19,27,28,46,29,30,34,47)(23,36,38,45,39,40,37,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,c^b=ch,d^b=df,e^b=eghi,g^b=gi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=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, 58665);