#9713682

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

3821 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,23,27)(19,26,22,24)(20,21,25,32)(28,29,30,31)(36,44)(39,46)(43,47)(45,48);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,19,23,22)(18,24,27,26)(20,28,25,30)(21,31,32,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,7)(3,8)(9,13)(11,16)(17,19,23,22)(18,24,27,26)(20,30,25,28)(21,29,32,31)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,20,23,25)(18,21,27,32)(19,28,22,30)(24,31,26,29)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21)(18,25)(19,29)(20,27)(22,31)(23,32)(24,28)(26,30)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,22,23,19)(18,26,27,24)(20,30,25,28)(21,29,32,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=g,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, 9620878);