#9481382

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388578);