#10195640

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

4066 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,e^2=gh,f^2=ghi,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, 10102836);