#9909984

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

4113 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,27)(20,26)(21,24)(22,30)(23,25)(28,29)(31,32)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,21)(18,24,25,27)(20,28,30,31)(22,29,26,32)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,7)(6,13)(8,11)(14,15)(18,25)(20,30)(24,27)(28,31)(34,41)(35,45)(36,47)(43,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,26)(19,28)(21,31)(22,25)(23,30)(24,32)(27,29)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,23,19)(18,24,25,27)(20,31,30,28)(22,29,26,32)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,26)(18,20,25,30)(19,29,21,32)(24,28,27,31)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)

Polycyclic presentation:

2^9|b^2=h,e^2=g,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, 9817180);