#9910706

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3735 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)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(33,34)(35,36)(37,43)(38,51)(39,40)(41,42)(44,52)(45,47)(46,55)(48,57)(49,53)(50,62)(54,56)(58,61)(59,63)(60,64);(2,6)(7,13)(8,11)(14,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(34,40)(35,42)(37,49)(44,51)(46,57)(47,54)(50,61)(60,63);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,35)(34,41)(36,40)(37,45)(38,46)(39,42)(43,54)(44,55)(47,53)(48,51)(49,56)(50,59)(52,57)(58,63)(60,62)(61,64);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(33,36,39,41)(34,42,40,35)(37,47,49,54)(38,48,52,55)(43,56,53,45)(44,57,51,46)(50,60,61,63)(58,64,62,59);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,21,23,18)(19,24,20,26)(22,25,30,29)(27,32,28,31)(33,37,39,49)(34,43,40,53)(35,45,42,56)(36,47,41,54)(38,50,52,61)(44,58,51,62)(46,59,57,64)(48,60,55,63);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)(33,38,39,52)(34,44,40,51)(35,46,42,57)(36,48,41,55)(37,50,49,61)(43,58,53,62)(45,59,56,64)(47,60,54,63)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=g,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, 9817902);