#8975578

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	#50598	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,e^2=gh,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, 8882774);