#9015306

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	#56296	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=hi,e^2=i,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, 8922502);