#9938221

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

4283 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)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(3,9)(8,15)(10,16)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(3,9)(8,15)(10,16)(18,23)(19,26)(24,31)(27,28)(34,36)(39,41)(40,44)(46,47);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,3)(4,11)(6,14)(7,9)(8,16)(10,15)(12,13)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845417);