#9243630

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

4194 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34)(35,43)(36,45)(37,42)(38,46)(39,41)(40,44)(47,48);(2,6)(7,9)(8,13)(14,15)(20,27)(24,30)(26,31)(28,32)(33,35,39,44)(34,40,41,43)(36,37,46,47)(38,42,45,48);(4,12)(8,13)(10,16)(14,15)(18,23)(19,25)(24,30)(26,31)(34,41)(35,44)(36,46)(42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,38)(35,37)(39,46)(40,42)(41,45)(43,48)(44,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,37)(34,42)(35,36)(38,40)(39,47)(41,48)(43,45)(44,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38)(34,36)(35,42)(37,40)(39,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150826);