#8119895

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	#38004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027091);