#8712357

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

4286 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)(35,44)(36,48)(37,46)(38,42)(39,43)(40,47)(41,45);(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)(34,40)(36,37)(38,46)(39,47)(41,48)(42,45)(43,44);(4,13)(8,12)(11,14)(15,16)(19,28)(20,29)(24,30)(25,32)(33,36,39,42)(34,38,43,48)(35,37,47,45)(40,46,44,41);(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)(34,38)(35,45)(37,47)(39,42)(40,41)(43,48)(44,46);(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)(33,37,39,45)(34,41,43,46)(35,36,47,42)(38,40,48,44);(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)(34,42)(35,46)(36,43)(37,44)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=i,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, 8619553);