#8115455

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

3743 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(18,22)(19,25)(20,28)(30,32)(33,34,39,37)(35,40,46,44)(36,47,50,41)(38,51,52,42)(43,55,48,53)(45,58,59,54)(49,56,62,60)(57,63,61,64);(2,7)(3,8)(9,14)(10,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(34,37)(40,44)(41,47)(42,51)(53,55)(54,58)(56,60)(63,64);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,39,46)(34,40,37,44)(36,48,50,43)(38,45,52,59)(41,55,47,53)(42,54,51,58)(49,61,62,57)(56,64,60,63);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36)(34,41)(35,43)(37,47)(38,49)(39,50)(40,53)(42,56)(44,55)(45,57)(46,48)(51,60)(52,62)(54,63)(58,64)(59,61);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,11,12,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,34)(35,44,46,40)(36,47,50,41)(38,51,52,42)(43,55,48,53)(45,58,59,54)(49,60,62,56)(57,64,61,63);(33,38,39,52)(34,42,37,51)(35,45,46,59)(36,49,50,62)(40,54,44,58)(41,56,47,60)(43,57,48,61)(53,63,55,64)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=gh,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, 8022651);