#8535379

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

4254 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34)(35,40)(36,45)(37,41)(38,44)(39,43)(42,48)(46,47);(2,7)(3,10)(8,15)(11,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,39,46)(34,40,43,47)(36,42,37,44)(38,45,48,41);(2,7)(4,9)(8,15)(12,16)(20,27)(23,25)(28,32)(30,31)(33,36,39,37)(34,41,43,45)(35,44,46,42)(38,40,48,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,37,39,36)(34,41,43,45)(35,42,46,44)(38,40,48,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,38)(34,42)(35,45)(36,47)(37,40)(39,48)(41,46)(43,44);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,34)(35,40)(36,41)(37,45)(38,42)(39,43)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8442575);