#8372152

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8279348);