#151349

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

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

normal subgroups only

This group has 390 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,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, 58545);