#247039

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	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 977 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)(3,13)(4,17)(5,8)(6,11)(7,12)(9,15)(10,18)(14,23)(16,22)(19,26)(20,25)(21,24)(27,28)(29,31)(30,32)(33,34,38,43)(35,40,46,56)(36,47,48,57)(37,39,50,42)(41,52,58,49)(44,59,60,63)(45,54,51,53)(55,62,64,61);(2,8)(5,6)(9,13)(10,17)(11,12)(15,21)(18,20)(22,24)(23,27)(25,26)(28,29)(31,32)(34,39)(36,48)(37,50)(40,54)(41,57)(42,43)(44,60)(45,51)(47,58)(53,56)(55,63)(59,64);(1,3,7,16)(2,9,12,24)(4,14,19,30)(5,15,6,21)(8,22,11,13)(10,23,25,32)(17,31,26,27)(18,28,20,29)(33,35,38,46)(34,40,43,56)(36,44,48,60)(37,51,50,45)(39,53,42,54)(41,55,58,64)(47,63,57,59)(49,62,52,61);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,17)(9,23)(11,26)(12,25)(13,27)(15,29)(16,30)(21,28)(22,31)(24,32)(33,36,38,48)(34,41,43,58)(35,44,46,60)(37,52,50,49)(39,47,42,57)(40,55,56,64)(45,62,51,61)(53,63,54,59)

Polycyclic presentation:

2^9|a^2=i,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, 154235);