#150254

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1347
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂	C₂×#1347
prime radical	#33604	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,15,18,46)(4,11)(5,24,26,57)(6,30)(7,32)(8,34)(9,14)(10,17)(12,27,42,28)(13,23)(16,45,35,47)(19,51,20,52)(21,53,29,59)(22,44)(25,43)(31,62,33,41)(36,48)(37,60)(38,58)(39,50)(40,55,63,56)(49,64)(54,61);(1,3)(2,10)(4,16)(5,25)(6,31)(7,33)(8,35)(9,18)(11,36)(12,41)(13,43)(14,39)(15,24)(17,28)(19,22)(20,23)(21,54)(26,49)(27,50)(29,60)(30,61)(32,37)(34,38)(40,52)(42,62)(44,64)(45,53)(46,57)(47,59)(48,56)(51,63)(55,58);(1,4,9,8)(2,11,14,34)(3,16,18,35)(5,21,26,29)(6,22,7,23)(10,36,39,38)(12,40,42,63)(13,30,44,32)(15,45,46,47)(17,48,50,58)(19,33,20,31)(24,53,57,59)(25,54,49,60)(27,55,28,56)(37,43,61,64)(41,52,62,51)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=ghi,h^a=hi,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57450);