#8602777

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=gh,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, 8509973);