#9815221

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,32)(28,30)(33,34,38,43)(35,39,46,40)(36,41,47,45)(37,44,48,42);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(2,7)(3,10)(8,13)(11,16)(18,20)(24,27)(25,30)(26,31)(33,35,38,46)(34,40,43,39)(36,44,47,42)(37,45,48,41);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,48)(42,46)(43,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,37,38,48)(34,42,43,44)(35,45,46,41)(36,39,47,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9722417);