#9015272

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	#56311	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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)(3,8)(4,7)(5,9)(6,10)(11,13)(12,15)(14,16)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,27,28,22)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3)(2,4)(5,11)(6,7)(8,10)(9,12)(13,16)(14,15)(17,19,23,31)(18,24,29,32)(20,28,30,21)(22,25,27,26)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,4,6,8)(2,7,10,3)(5,13,14,12)(9,11,16,15)(17,20,23,30)(18,25,29,26)(19,21,31,28)(22,32,27,24)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,4)(2,3)(5,12)(6,8)(7,10)(9,11)(13,14)(15,16)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,8)(4,10)(5,11)(6,7)(9,13)(12,16)(14,15)(17,21,23,28)(18,27,29,22)(19,30,31,20)(24,25,32,26)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22,23,27)(18,28,29,21)(19,25,31,26)(20,24,30,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=hi,e^2=i,f^2=h,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, 8922468);