#9508850

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	#41148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(33,34)(35,42)(36,44)(37,41)(38,39)(40,45)(43,47)(46,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35,39,41)(34,37,38,42)(36,43,45,48)(40,47,44,46);(2,7)(6,13)(8,11)(14,15)(18,23)(20,27)(21,30)(26,28)(34,38)(36,45)(37,42)(43,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,36,39,45)(34,40,38,44)(35,43,41,48)(37,47,42,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,42)(34,41,38,35)(36,46,45,47)(40,43,44,48);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,38,39,34)(35,42,41,37)(36,44,45,40)(43,46,48,47)

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9416046);