#9243494

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3715 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(2,5)(4,12)(7,9)(10,13)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,3,6,11)(2,7,5,9)(4,13,12,10)(8,16,14,15)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,13)(14,16)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,5,6,2)(3,9,11,7)(4,14,12,8)(10,16,13,15)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,f^2=g,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, 9150690);