#9911004

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3766 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=h,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^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818200);