#9713341

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620537);