#8720648

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,11)(4,13)(5,8)(6,12)(7,9)(10,15)(14,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,10,16,13)(18,24)(19,20)(26,29)(27,28)(35,41)(38,44)(42,47)(45,48);(2,9)(4,15)(8,11)(12,14)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,10)(3,12)(5,14)(6,8)(7,15)(9,13)(11,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,10,16,13)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,6,7,16)(2,12,9,14)(3,10,5,13)(4,11,15,8)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=i,e^2=g,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, 8627844);