#9015331

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	#56303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922527);