#9508908

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	#50594	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,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, 9416104);