#8994662

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	#52335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(4,10)(5,6)(7,12)(9,13)(11,16)(14,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(1,3,7,14)(2,8,12,15)(4,11,13,5)(6,10,16,9)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(1,4,7,13)(2,9,12,10)(3,5,14,11)(6,8,16,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4)(2,10)(3,11)(5,14)(6,15)(7,13)(8,16)(9,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5,7,11)(2,6,12,16)(3,13,14,4)(8,9,15,10)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,6)(2,11)(3,10)(4,15)(5,12)(7,16)(8,13)(9,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=hi,e^2=gh,f^2=g,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, 8901858);