#8712363

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

4237 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)(3,12)(4,14)(5,13)(6,8)(7,9)(10,15)(11,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34)(35,41)(36,39)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3,7,6)(2,8,9,12)(4,11,15,13)(5,10,16,14)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,37)(36,44)(38,42)(39,45)(40,47)(41,43)(46,48);(2,9)(4,15)(8,12)(11,13)(19,28)(20,29)(24,30)(25,32)(33,36,38,46)(34,39,43,40)(35,44,42,48)(37,45,41,47);(1,4,7,15)(2,10,9,14)(3,13,6,11)(5,12,16,8)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,36)(34,40)(35,44)(37,47)(38,46)(39,43)(41,45)(42,48);(1,5)(2,11)(3,10)(4,12)(6,14)(7,16)(8,15)(9,13)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,35)(34,41)(36,44)(37,43)(38,42)(39,47)(40,45)(46,48);(1,6,7,3)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,37,38,41)(34,42,43,35)(36,45,46,47)(39,48,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,f^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, 8619559);