#9267017

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3710 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,7)(4,12)(5,8)(11,16)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32)(34,35)(39,41)(40,42)(46,47);(2,7)(3,10)(4,12)(13,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,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, 9174213);