#8119379

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026575);