#8417013

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

3713 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,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, 8324209);