#10519394

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,7)(4,14)(5,9)(6,10)(8,13)(11,16)(12,15)(19,22)(24,26)(27,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(3,11)(4,13)(7,16)(8,14)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34,38,42)(35,39,44,47)(36,40,43,46)(37,41,45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,9)(10,16)(14,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,38,44)(34,39,42,47)(36,37,43,45)(40,41,46,48);(1,4,6,13)(2,8,10,14)(3,12,11,5)(7,9,16,15)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,12)(2,9,10,15)(3,13,11,4)(7,8,16,14)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,36)(34,40)(35,37)(38,43)(39,41)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,15)(6,10)(9,12)(11,16)(13,14)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10426590);