#9245124

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,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, 9152320);