#8898532

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	#48454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8805728);