#8709120

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
prime radical	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,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, 8616316);