#9910204

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

4217 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)(3,4)(5,9)(6,8)(7,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32)(35,44)(37,47)(39,40)(42,45);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,30,32,29)(33,34,38,43)(35,40,44,39)(36,41,46,48)(37,42,47,45);(2,8)(4,7)(9,16)(12,15)(17,19,23,20)(18,21,26,24)(22,27,31,28)(25,29,32,30)(33,35,38,44)(34,39,43,40)(36,45,46,42)(37,41,47,48);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,16,13)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,30,32,29)(33,35)(34,40)(36,45)(37,41)(38,44)(39,43)(42,46)(47,48);(1,5,6,14)(2,9,8,16)(3,10,11,13)(4,12,7,15)(17,21,23,24)(18,19,26,20)(22,30,31,29)(25,28,32,27)(33,36)(34,41)(35,45)(37,40)(38,46)(39,47)(42,44)(43,48);(1,3)(2,7)(4,8)(5,13)(6,11)(9,12)(10,14)(15,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,45)(44,48)

Polycyclic presentation:

2^9|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, 9817400);