#8757443

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

3709 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664639);