#151250

order	512
nilpotency class	5
isoclinism	isoclinic class of #151162
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 276 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)(11,19,22,36)(12,16,24,32)(13,25,26,17)(15,28,30,20)(21,34,39,35)(23,33,27,31)(29,38,40,37);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,29)(18,30)(23,37)(24,28)(25,40)(26,39)(27,38)(31,34)(32,36)(33,35);(1,3,5,4)(2,6,7,8)(12,23)(13,26)(16,31)(17,25)(19,34)(20,37)(21,39)(24,27)(28,38)(29,40)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,c^2=f,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=ehi,g^c=gi,e^d=eh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58446);