#9267019

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	#42064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3840 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=gi,e^2=h,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, 9174215);