#10195511

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	#55525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gh,e^2=g,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, 10102707);