#10142131

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	#51084	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=hi,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, 10049327);