#9910771

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	#49854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=h,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, 9817967);