#9912880

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

4011 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820076);