#8406456

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

3701 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8313652);