#9481532

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3641 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gi,e^2=gh,f^2=g,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, 9388728);