#10128052

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

4021 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)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34,38,39)(35,40,43,46)(36,41,47,44)(37,45,48,42);(4,11)(8,14)(10,15)(12,16)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(33,34,38,39)(35,40,43,46)(36,44,47,41)(37,42,48,45);(2,7)(3,9)(8,14)(10,15)(18,25)(20,30)(24,26)(27,31)(34,39)(35,43)(41,44)(42,45);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,35)(34,40)(36,45)(37,44)(38,43)(39,46)(41,48)(42,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,36)(34,41)(35,42)(37,40)(38,47)(39,44)(43,45)(46,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,37)(34,42)(35,44)(36,46)(38,48)(39,45)(40,47)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=hi,f^2=g,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, 10035248);