#247040

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 937 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,15)(4,20)(5,22)(6,13)(7,12)(8,14)(9,17)(10,21)(11,23)(16,28)(18,30)(19,29)(24,27)(25,32)(26,31)(33,34,39,44)(35,45,36,41)(37,50,52,61)(38,53,55,43)(40,49,57,47)(42,54,60,51)(46,56,48,59)(58,63,62,64);(1,3,8,19)(2,9,14,30)(4,5,16,24)(6,18,7,17)(10,22,31,27)(11,28,32,20)(12,29,13,15)(21,25,26,23)(33,35,39,36)(34,40,44,57)(37,48,52,46)(38,49,55,47)(41,53,45,43)(42,59,60,56)(50,58,61,62)(51,63,54,64);(1,4,8,16)(2,10,14,31)(3,5,19,24)(6,21,7,26)(9,27,30,22)(11,29,32,15)(12,20,13,28)(17,23,18,25)(33,36,39,35)(34,41,44,45)(37,48,52,46)(38,47,55,49)(40,43,57,53)(42,58,60,62)(50,56,61,59)(51,63,54,64);(1,5)(2,11)(3,16)(4,19)(6,25)(7,23)(8,24)(9,28)(10,29)(12,27)(13,22)(14,32)(15,31)(17,21)(18,26)(20,30)(33,37,39,52)(34,42,44,60)(35,46,36,48)(38,54,55,51)(40,56,57,59)(41,58,45,62)(43,61,53,50)(47,64,49,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154236);