#8761988

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669184);