#8107338

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	#41304	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(19,25)(22,28)(26,31)(29,32)(34,40)(36,44)(38,41)(43,45);(1,3,7,12)(2,8,9,15)(4,6,10,13)(5,14,16,11)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,40)(35,41,37,38)(36,42,44,47)(43,48,45,46);(2,9)(3,12)(4,10)(11,14)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35,39,37)(34,41,40,38)(36,45,44,43)(42,46,47,48);(1,4,7,10)(2,5,9,16)(3,13,12,6)(8,11,15,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,39,44)(34,42,40,47)(35,43,37,45)(38,46,41,48);(1,5)(2,10)(3,14)(4,9)(6,15)(7,16)(8,13)(11,12)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,37,39,35)(34,38,40,41)(36,45,44,43)(42,46,47,48);(1,6,7,13)(2,11,9,14)(3,10,12,4)(5,15,16,8)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38)(34,35)(36,46)(37,40)(39,41)(42,43)(44,48)(45,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=g,f^2=h,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, 8014534);