#8107589

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	#44426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(18,22)(20,28)(24,26)(27,29)(34,37)(35,41)(36,44)(46,48);(2,7)(3,10)(8,14)(11,16)(17,18,23,22)(19,24,21,26)(20,25,28,30)(27,31,29,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,23,21)(18,24,22,26)(20,29,28,27)(25,32,30,31)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,35,38,41)(34,39,37,42)(36,43,44,47)(40,46,45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,37,38,34)(35,42,41,39)(36,45,44,40)(43,48,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014785);