#8898910

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	#35436	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4116 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,f^2=h,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, 8806106);