#8402594

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	#59548	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(21,25,31,28)(22,26,32,30)(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)(17,19,23,27)(18,24,20,29)(21,30,31,26)(22,28,32,25)(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)(22,32)(24,29)(26,30)(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)(21,25,31,28)(22,30,32,26)(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)(19,26)(20,28)(22,29)(23,31)(24,32)(27,30)(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)(17,22,23,32)(18,26,20,30)(19,28,27,25)(21,29,31,24)(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=i,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, 8309790);