#8926624

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

3703 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:

(3,9)(7,10)(11,15)(14,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(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);(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(18,23)(19,26)(25,28)(27,32)(35,44)(40,43)(45,58)(46,51)(53,57)(54,56)(59,61)(63,64);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(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);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(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);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(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);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(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=gi,b^2=hi,c^2=g,d^2=gh,e^2=gh,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, 8833820);