#8115570

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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)(9,14)(10,15)(11,16)(17,18,22,26)(19,25,21,23)(20,24,29,32)(27,31,28,30)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30)(34,38)(35,42)(40,44)(43,48);(2,7)(3,8)(9,14)(10,15)(17,18,22,26)(19,23,21,25)(20,24,29,32)(27,30,28,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,11,12,16)(17,21,22,19)(18,25,26,23)(20,28,29,27)(24,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022766);