#10196063

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	#41821	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10103259);