#9267028

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	#42906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=i,e^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, 9174224);