#8159643

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066839);