#10142136

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	#40420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=ghi,e^2=gh,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, 10049332);