#9039121

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

3703 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(19,22)(24,26)(27,29)(31,32)(33,34,39,43)(35,37,41,50)(36,46,49,40)(38,42,54,58)(44,47,56,51)(45,52,53,57)(48,60,63,55)(59,61,62,64);(2,7)(4,8)(9,13)(12,15)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(35,41)(37,50)(44,56)(45,53)(47,51)(52,57)(59,62)(61,64);(1,2,6,7)(3,8,11,4)(5,9,14,13)(10,15,16,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,37)(36,44)(38,53)(39,41)(40,51)(42,57)(43,50)(45,54)(46,47)(48,62)(49,56)(52,58)(55,61)(59,63)(60,64);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36,39,49)(34,40,43,46)(35,44,41,56)(37,51,50,47)(38,48,54,63)(42,55,58,60)(45,59,53,62)(52,64,57,61);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,41)(35,43)(36,47)(38,52)(39,50)(40,44)(42,53)(45,58)(46,56)(48,61)(49,51)(54,57)(55,59)(60,62)(63,64);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38)(34,42)(35,45)(36,48)(37,52)(39,54)(40,55)(41,53)(43,58)(44,59)(46,60)(47,61)(49,63)(50,57)(51,64)(56,62)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946317);