#201064

order	512
nilpotency class	3
isoclinism	isoclinic class of #200938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

993 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi

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, 108260);