#8718168

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

4184 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)(19,27)(20,29)(21,24)(22,28)(23,25)(26,31)(30,32)(33,34,38,39)(35,36,42,41)(37,40,47,48)(43,44,45,46);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,26,32,29)(33,34,38,39)(35,41,42,36)(37,40,47,48)(43,46,45,44);(4,13)(8,12)(11,14)(15,16)(18,25)(20,31)(24,27)(28,30)(34,39)(35,42)(40,48)(43,45);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,20)(18,26)(19,28)(21,30)(22,24)(23,31)(25,29)(27,32)(33,35)(34,36)(37,45)(38,42)(39,41)(40,46)(43,47)(44,48);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,26,32,29)(33,36)(34,35)(37,46)(38,41)(39,42)(40,45)(43,48)(44,47);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22,23,32)(18,28,25,30)(19,26,21,29)(20,27,31,24)(33,37,38,47)(34,40,39,48)(35,43,42,45)(36,44,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=ghi,e^2=g,f^2=h,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, 8625364);