#9910205

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

4243 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^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, 9817401);