#9850759

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9757955);