#8720126

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=i,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, 8627322);