#10246863

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	#55497	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154059);