#150606

order	512
nilpotency class	5
isoclinism	isoclinic class of #150605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11801
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=hi,d^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efgi,g^b=gi,e^c=ei,e^d=eh,f^e=fi

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, 57802);