#8402426

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	#59228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309622);