#8734343

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	#57479	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,32)(30,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,21)(18,24,25,27)(20,31,28,29)(22,30,26,32)(33,35,38,43)(34,39,42,48)(36,37,44,47)(40,46,45,41);(2,3)(7,9)(8,10)(15,16)(18,25)(20,28)(24,27)(29,31)(35,43)(36,44)(39,48)(40,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,26)(19,29)(21,31)(22,25)(23,28)(24,30)(27,32)(33,36)(34,40)(35,37)(38,44)(39,46)(41,48)(42,45)(43,47);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,30,26,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,48)(44,45);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,29,27,31)(33,37)(34,41)(35,44)(36,43)(38,47)(39,40)(42,46)(45,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641539);