#8926525

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

3706 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)(3,10,5,8)(4,6,11,9)(13,16,15,14)(18,22)(20,27)(23,28)(26,31)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,57)(46,51)(47,53)(48,58)(49,59)(50,55)(52,60)(54,56)(61,63)(62,64);(1,3,7,5)(2,8,12,10)(4,15,11,13)(6,16,9,14)(19,25)(23,28)(26,31)(30,32)(34,40)(37,51)(41,53)(42,55)(44,56)(48,52)(57,64)(59,61);(1,4,7,11)(2,9,12,6)(3,13,5,15)(8,16,10,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,41)(36,45)(37,44)(38,43)(39,47)(40,53)(42,57)(46,54)(48,61)(49,60)(50,62)(51,56)(52,59)(55,64)(58,63);(1,5,7,3)(2,10,12,8)(4,15,11,13)(6,16,9,14)(33,36,39,50)(34,42,40,55)(35,45,47,62)(37,52,51,48)(38,49,54,63)(41,57,53,64)(43,60,46,58)(44,59,56,61);(1,6,7,9)(2,11,12,4)(3,14,5,16)(8,15,10,13)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,51)(34,43,40,46)(35,44,47,56)(36,48,50,52)(38,53,54,41)(42,58,55,60)(45,61,62,59)(49,57,63,64);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,44)(35,46)(36,49)(37,53)(39,54)(40,56)(41,51)(42,59)(43,47)(45,58)(48,57)(50,63)(52,64)(55,61)(60,62)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^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, 8833721);