#8107584

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	#42104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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)(3,12)(4,5)(6,13)(7,11)(8,15)(9,16)(10,14)(18,23)(19,28)(20,30)(21,31)(35,37)(38,48)(40,44)(42,45);(1,3)(2,8)(4,13)(5,14)(6,9)(7,15)(10,16)(11,12)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34,39,43)(35,44,37,40)(36,46,47,41)(38,42,48,45);(1,4,7,16)(2,5,11,9)(3,13,15,10)(6,8,14,12)(17,19,22,28)(18,24,23,25)(20,27,30,32)(21,29,31,26)(33,35,39,37)(34,40,43,44)(36,42,47,45)(38,41,48,46);(1,4)(2,9)(3,13)(5,11)(6,8)(7,16)(10,15)(12,14)(17,19)(18,25)(20,27)(21,29)(22,28)(23,24)(26,31)(30,32)(33,36,39,47)(34,41,43,46)(35,45,37,42)(38,44,48,40);(1,5,7,9)(2,4,11,16)(3,14,15,6)(8,13,12,10)(17,20)(18,26)(19,27)(21,24)(22,30)(23,29)(25,31)(28,32)(33,37,39,35)(34,40,43,44)(36,42,47,45)(38,46,48,41);(1,6,7,14)(2,10,11,13)(3,9,15,5)(4,8,16,12)(17,21,22,31)(18,27,23,32)(19,29,28,26)(20,24,30,25)(33,38)(34,42)(35,41)(36,44)(37,46)(39,48)(40,47)(43,45)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,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, 8014780);