#10247206

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

3716 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)(3,13)(4,9)(5,15)(6,10)(7,11)(8,12)(14,16)(33,34,39,40)(35,36,44,41)(37,42,49,53)(38,51,52,43)(45,47,57,54)(46,55,58,48)(50,62,61,56)(59,63,64,60);(1,3)(2,7)(4,14)(5,6)(8,11)(9,10)(12,13)(15,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,40)(35,44)(37,49)(43,51)(46,58)(47,54)(50,61)(60,63);(1,4,8,5)(2,9,12,15)(3,14,11,6)(7,10,13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,39,44)(34,41,40,36)(37,45,49,57)(38,46,52,58)(42,54,53,47)(43,55,51,48)(50,59,61,64)(56,63,62,60);(1,5,8,4)(2,9,12,15)(3,14,11,6)(7,16,13,10)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,41)(34,35,40,44)(37,47,49,54)(38,48,52,55)(42,45,53,57)(43,46,51,58)(50,60,61,63)(56,59,62,64);(1,6,8,14)(2,10,12,16)(3,5,11,4)(7,9,13,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,37)(34,42)(35,45)(36,47)(38,50)(39,49)(40,53)(41,54)(43,56)(44,57)(46,59)(48,60)(51,62)(52,61)(55,63)(58,64);(1,7,8,13)(2,11,12,3)(4,10,5,16)(6,15,14,9)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,52)(34,43,40,51)(35,46,44,58)(36,48,41,55)(37,50,49,61)(42,56,53,62)(45,59,57,64)(47,60,54,63)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154402);