#9912866

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

3875 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,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, 9820062);