#8712452

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

4219 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,e^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, 8619648);