#8115368

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3735 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022564);