#150422

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efg,f^b=fi,d^c=dh,f^c=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57618);