#150257

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

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

normal subgroups only

This group has 301 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ghi,e^2=i,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, 57453);