#10195871

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34,38,39)(35,43,44,40)(36,41,45,48)(37,46,47,42);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,24)(20,29)(21,30)(22,25)(23,28)(26,32)(27,31)(34,39)(36,45)(40,43)(42,46);(2,7)(3,10)(8,15)(11,12)(18,25)(19,28)(20,30)(27,32)(33,35,38,44)(34,40,39,43)(36,42,45,46)(37,41,47,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,30)(18,26,25,31)(19,29,28,21)(22,27,24,32)(33,35)(34,40)(36,46)(37,41)(38,44)(39,43)(42,45)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,21,23,29)(18,27,25,32)(19,20,28,30)(22,31,24,26)(33,36)(34,41)(35,42)(37,43)(38,45)(39,48)(40,47)(44,46);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,28)(19,25)(20,27)(21,26)(23,24)(29,31)(30,32)(33,37,38,47)(34,42,39,46)(35,41,44,48)(36,40,45,43)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=hi,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, 10103067);