#8602629

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

3961 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,39,42)(35,40,45,38)(36,46,37,41)(43,47,44,48);(2,7)(3,11)(4,14)(13,15)(18,24)(19,20)(26,31)(27,28)(33,35,39,45)(34,40,42,38)(36,44,37,43)(41,47,46,48);(2,7)(4,14)(8,10)(12,16)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,36,39,37)(34,41,42,46)(35,43,45,44)(38,47,40,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,37,39,36)(34,41,42,46)(35,44,45,43)(38,47,40,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,34)(35,38)(36,46)(37,41)(39,42)(40,45)(43,48)(44,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,43)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,f^2=g,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, 8509825);