#8824433

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	#48720	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(18,22)(20,27)(23,28)(26,31)(34,38)(35,41)(40,43)(42,47);(1,3,5,11)(2,6,9,8)(4,10,13,16)(7,14,12,15)(19,25)(23,28)(26,31)(30,32)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(1,2,5,9)(3,8,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,13)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=gi,e^2=ghi,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, 8731629);