#8656217

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=gi,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, 8563413);