#8734313

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 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)(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)(29,31)(30,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,48)(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)(20,30,31,28)(22,29,32,26)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,3)(7,9)(8,10)(15,16)(18,25)(20,31)(24,27)(28,30)(36,44)(40,48)(42,46)(45,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,26)(19,28)(21,30)(22,27)(23,31)(24,32)(25,29)(33,36,38,44)(34,40,41,48)(35,42,43,46)(37,45,39,47);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,32,26)(33,37,38,39)(34,35,41,43)(36,45,44,47)(40,42,48,46);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,22)(18,28)(19,29)(20,24)(21,26)(23,32)(25,30)(27,31)(33,35)(34,37)(36,46)(38,43)(39,41)(40,47)(42,44)(45,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^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, 8641509);