#10106307

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=g,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, 10013503);