#8943952

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851148);