#8295607

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	#41290	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18)(19,27)(20,30)(21,24)(22,28)(23,25)(26,31)(29,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,30,32,26)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(5,13)(9,12)(10,16)(14,15)(18,25)(22,32)(24,27)(26,30)(34,38)(39,41)(40,44)(47,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,24,32,27)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,19)(18,27,25,24)(20,29,31,28)(22,26,32,30)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,24,31,27)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202803);