#9910714

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3741 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:

(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(19,25)(23,29)(26,31)(30,32)(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);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(18,22)(23,29)(24,27)(30,32)(34,40)(35,45)(37,42)(43,53)(46,49)(48,59)(52,57)(60,62);(1,3,7,4)(2,9,5,8)(6,11,14,12)(10,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(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);(1,4,7,3)(2,8,5,9)(6,12,14,11)(10,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(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);(1,5,7,2)(3,8,4,9)(6,10,14,13)(11,16,12,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(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);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(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|a^2=gh,b^2=gh,c^2=gh,d^2=gh,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, 9817910);