#8125476

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

4178 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032672);