#9782690

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3891 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,24,26,21)(20,22,29,32)(27,28,31,30)(33,34)(35,37)(36,41)(38,43)(39,44)(40,42)(45,46)(47,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,26)(18,24,25,21)(20,27,29,31)(22,28,32,30)(33,35,39,42)(34,40,44,37)(36,43,46,47)(38,41,48,45);(2,7)(6,13)(8,11)(14,15)(18,25)(19,26)(20,29)(28,30)(33,36,39,46)(34,41,44,45)(35,43,42,47)(37,38,40,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20)(18,22)(19,27)(21,30)(23,29)(24,28)(25,32)(26,31)(33,37,39,40)(34,42,44,35)(36,38,46,48)(41,47,45,43);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,30)(22,27)(23,24)(25,26)(28,29)(31,32)(33,38)(34,43)(35,45)(36,40)(37,46)(39,48)(41,42)(44,47);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22)(18,20)(19,28)(21,31)(23,32)(24,27)(25,29)(26,30)(33,34)(35,40)(36,41)(37,42)(38,47)(39,44)(43,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9689886);