#8926617

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833813);