#9509008

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	#38314	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=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, 9416204);