#8602767

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	#39813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509963);