#8634509

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	#46339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541705);