#8402263

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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)(3,12)(4,6)(5,15)(7,9)(8,16)(10,14)(11,13)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(1,3,7,16)(2,8,9,12)(4,13,14,5)(6,15,10,11)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,9)(6,10)(8,12)(11,15)(18,23)(21,25)(24,26)(28,32)(33,35,38,36)(34,39,41,42)(37,44,45,43)(40,47,48,46);(1,4)(2,10)(3,13)(5,16)(6,9)(7,14)(8,11)(12,15)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,48,47);(1,5,7,13)(2,11,9,15)(3,14,16,4)(6,8,10,12)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,47)(45,48);(1,6,7,10)(2,4,9,14)(3,15,16,11)(5,8,13,12)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)(33,37,38,45)(34,40,41,48)(35,43,36,44)(39,46,42,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,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, 8309459);