#9716864

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	#2456	C₂²
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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(34,39)(35,42)(41,47)(43,44);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(2,8)(3,11)(4,14)(10,15)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,45,47,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,21,23,20)(18,26,27,25)(19,30,22,29)(24,32,28,31)(33,36,38,46)(34,41,39,47)(35,43,42,44)(37,45,40,48);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,37)(34,35)(36,45)(38,40)(39,42)(41,43)(44,47)(46,48);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 9624060);