#8824461

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

4243 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)(3,8)(4,5)(6,7)(9,12)(10,16)(11,13)(14,15)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27)(34,39)(37,40)(41,46)(45,47);(2,7)(5,15)(8,9)(11,16)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(36,44)(41,46)(43,48)(45,47);(1,3,6,12)(2,8,7,9)(4,13,14,10)(5,11,15,16)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3)(2,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,16)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,4)(2,5)(3,10)(6,14)(7,15)(8,16)(9,11)(12,13)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,5)(2,4)(3,11)(6,15)(7,14)(8,13)(9,10)(12,16)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=hi,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, 8731657);