#10247623

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

3731 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)(3,11)(4,9)(5,15)(6,7)(8,10)(12,16)(13,14)(33,34,39,37)(35,43,46,40)(36,47,48,41)(38,51,52,42)(44,53,49,55)(45,54,59,57)(50,56,61,60)(58,64,62,63);(1,3,8,6)(2,7,10,11)(4,12,5,13)(9,14,15,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,37)(36,48)(40,43)(42,51)(44,49)(50,61)(54,57)(58,62);(1,4,8,5)(2,9,10,15)(3,12,6,13)(7,14,11,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35)(34,40)(36,49)(37,43)(38,45)(39,46)(41,55)(42,54)(44,48)(47,53)(50,62)(51,57)(52,59)(56,64)(58,61)(60,63);(1,5,8,4)(2,9,10,15)(3,12,6,13)(7,16,11,14)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,39,48)(34,41,37,47)(35,44,46,49)(38,50,52,61)(40,53,43,55)(42,56,51,60)(45,58,59,62)(54,63,57,64);(1,6,8,3)(2,7,10,11)(4,13,5,12)(9,14,15,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,37,39,34)(35,43,46,40)(36,41,48,47)(38,42,52,51)(44,53,49,55)(45,54,59,57)(50,60,61,56)(58,64,62,63);(1,7)(2,6)(3,10)(4,14)(5,16)(8,11)(9,13)(12,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,52)(34,42,37,51)(35,45,46,59)(36,50,48,61)(40,54,43,57)(41,56,47,60)(44,58,49,62)(53,63,55,64)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,e^2=g,f^2=h,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, 10154819);