#8372129

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	#45663	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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)(19,24)(20,21)(22,25)(23,26)(27,31)(28,29)(30,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,2)(3,8)(4,5)(6,7)(9,14)(10,16)(11,13)(12,15)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31)(35,41)(38,45)(42,47)(46,48);(2,7)(5,9)(8,15)(11,16)(17,19,23,20)(18,24,26,21)(22,28,30,27)(25,29,32,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,4)(2,9)(3,10)(5,7)(6,14)(8,11)(12,13)(15,16)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,9)(2,4,7,14)(3,11,12,16)(8,13,15,10)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,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, 8279325);