#9938149

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

4123 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,39,41)(35,40,44,37)(36,47,46,42)(38,48,45,43);(2,6)(7,14)(8,11)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,44)(34,40,41,37)(36,38,46,45)(42,43,47,48);(2,6)(7,14)(8,11)(15,16)(18,22)(23,25)(24,28)(30,31)(34,41)(35,44)(36,46)(43,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,46)(34,42,41,47)(35,45,44,38)(37,43,40,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37)(34,35)(36,43)(38,47)(39,40)(41,44)(42,45)(46,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,45)(34,43,41,48)(35,46,44,36)(37,42,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=g,f^2=h,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, 9845345);