#9783141

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	#42096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^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, 9690337);