#9240980

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
prime radical	#43339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=gh,f^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, 9148176);