#10464532

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^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, 10371728);