#8712407

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

4241 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)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,39,42)(35,43,45,38)(36,46,48,40)(37,47,44,41);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,38)(36,37)(39,45)(40,47)(41,46)(42,43)(44,48);(4,13)(8,12)(11,14)(15,16)(19,28)(20,29)(24,30)(25,32)(33,36,39,48)(34,40,42,46)(35,37,45,44)(38,47,43,41);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,40)(35,44)(37,45)(38,41)(39,48)(42,46)(43,47);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,37,39,44)(34,41,42,47)(35,36,45,48)(38,46,43,40);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,38)(34,35)(36,47)(37,40)(39,43)(41,48)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619603);