#8551222

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

(1,2,7,6)(3,8,10,5)(4,9,13,14)(11,15,16,12)(18,24)(19,20)(21,29)(30,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,7,10)(2,8,6,5)(4,11,13,16)(9,15,14,12)(17,18)(19,25)(20,22)(21,26)(23,24)(27,32)(28,30)(29,31)(35,41)(38,44)(42,47)(45,48);(3,10)(5,8)(11,16)(12,15)(17,19,23,20)(18,25,24,22)(21,28,29,27)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,20,23,19)(18,25,24,22)(21,28,29,27)(26,32,31,30)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,21,23,29)(18,26,24,31)(19,27,20,28)(22,30,25,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,22,23,25)(18,20,24,19)(21,30,29,32)(26,28,31,27)(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=i,c^2=g,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458418);