#150205

order	512
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1344
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	C₂×#1344
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,15,18,46)(4,10,21,36)(5,22,24,42)(7,14,29,43)(8,13,28,44)(9,31,33,23)(11,37,39,26)(16,38,27,56)(17,45,48,41)(19,47,50,60)(20,40,51,63)(25,35,55,62)(30,52,57,64)(32,54,59,49)(34,58,61,53);(1,3)(2,9,13,34)(4,16)(5,23)(6,27)(7,30)(8,19)(10,32,14,45)(11,38,22,52)(12,41,44,59)(15,35,47,40)(17,25)(18,29)(20,49)(21,50)(24,54)(26,53)(28,57)(31,55)(33,43,61,36)(37,48)(39,46,42,60)(51,58)(56,63,64,62);(1,4,8,7)(2,10,13,14)(3,16,19,30)(5,20,26,25)(6,21,28,29)(9,32,34,45)(11,35,22,40)(12,36,44,43)(15,38,47,52)(17,23,49,53)(18,27,50,57)(24,51,37,55)(31,54,58,48)(33,59,61,41)(39,62,42,63)(46,56,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57401);