#9863961

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

4155 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)(3,10)(4,5)(6,8)(7,14)(9,11)(12,15)(13,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,10)(9,15)(11,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,34,38,41)(35,39,43,36)(37,40,47,45)(42,46,48,44);(2,8)(4,11)(7,10)(12,16)(18,25)(19,29)(20,31)(27,32)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,4,6,11)(2,9,8,5)(3,12,14,16)(7,15,10,13)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,3)(2,10)(4,12)(5,15)(6,14)(7,8)(9,13)(11,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5,6,9)(2,11,8,4)(3,13,14,15)(7,16,10,12)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=i,f^2=g,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, 9771157);