#8505550

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

4143 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,43)(35,44,46,40)(36,37,45,41)(38,42,48,47);(3,4)(5,13)(7,10)(8,15)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,46)(34,40,43,44)(36,38,45,48)(37,47,41,42);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,34,39,43)(35,40,46,44)(36,41,45,37)(38,42,48,47);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36)(34,41)(35,38)(37,43)(39,45)(40,42)(44,47)(46,48);(1,5,6,13)(2,8,9,15)(3,11,4,12)(7,14,10,16)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,41)(34,36,43,45)(35,42,46,47)(38,40,48,44);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38)(34,42)(35,45)(36,46)(37,40)(39,48)(41,44)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412746);