#150824

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	#28495	C₂
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

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

normal subgroups only

This group has 344 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)(3,15,18,49)(4,21,24,44)(5,27,25,41)(7,13,29,47)(8,14,34,46)(9,22,37,56)(10,30,11,33)(16,51,28,38)(17,39,54,45)(19,50,52,36)(20,42,32,57)(23,43,59,40)(26,58,31,61)(35,60,62,63)(48,53,64,55);(1,3)(2,9)(4,22)(5,28)(6,32)(7,19)(8,35)(10,36)(11,42)(12,45)(13,38)(14,48)(15,44)(16,26)(17,31)(18,34)(20,23)(21,57)(24,55)(25,53)(27,60)(29,62)(30,56)(33,54)(37,46)(39,43)(40,51)(41,49)(47,64)(50,61)(52,59)(58,63);(1,4,23,33,7,26,8,25)(2,10,14,21,13,41,43,58)(3,16,35,54,19,22,20,53)(5,29,24,34,30,6,31,59)(9,36,39,63,38,49,48,57)(11,40,44,12,27,46,61,47)(15,45,42,51,50,64,60,37)(17,18,56,62,55,52,28,32)

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,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, 58020);