#9015528

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
prime radical	#56384	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(17,18,23,21)(19,24,22,29)(20,25,27,30)(26,31,28,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,45)(44,47)(46,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,19,23,22)(18,24,21,29)(20,26,27,28)(25,31,30,32)(33,35)(34,36)(37,43)(38,39)(40,42)(41,44)(45,48)(46,47);(2,8)(4,7)(5,14)(11,16)(18,21)(19,22)(20,27)(31,32)(33,36,38,40)(34,39,42,35)(37,45,46,44)(41,43,48,47);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20)(18,25)(19,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,46)(47,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,21,23,18)(19,24,22,29)(20,25,27,30)(26,32,28,31)(33,35)(34,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,22,23,19)(18,24,21,29)(20,28,27,26)(25,31,30,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8922724);