#10316825

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10224021);