#150828

order	512
nilpotency class	5
isoclinism	isoclinic class of #150824
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11877
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,8,13,23,48)(3,15,18,35,20,52,53,39)(4,21,24,50,25,59,32,46)(5,27,26,42,30,11,43,10)(7,14,33,41,29,40,62,45)(9,16,36,28,37,22,60,56)(17,38,54,47,55,49,61,51)(19,44,57,58,34,63,31,64);(1,3,8,20)(2,9,13,37)(4,22,25,16)(5,28,30,56)(6,31,23,57)(7,34,29,19)(10,39,42,35)(11,44,27,63)(12,47,48,51)(14,49,40,38)(15,50,52,46)(17,32,55,24)(18,33,53,62)(21,58,59,64)(26,54,43,61)(36,45,60,41);(1,4,7,26,8,25,29,43)(2,10,40,59,13,42,14,21)(3,16,19,54,20,22,34,61)(5,23,24,62,30,6,32,33)(9,35,49,64,37,39,38,58)(11,45,50,48,27,41,46,12)(15,51,63,60,52,47,44,36)(17,53,28,31,55,18,56,57)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=eh,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, 58024);