#8893690

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	#46906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,7,10,14)(4,5,8,13)(11,12,15,16)(17,18,22,23)(19,24,27,31)(20,21,30,28)(25,32,29,26)(33,34)(35,42)(36,45)(37,41)(38,39)(40,44)(43,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,18,22,23)(19,24,27,31)(20,28,30,21)(25,26,29,32)(33,35)(34,40)(36,43)(37,46)(38,42)(39,44)(41,47)(45,48);(3,10)(7,14)(11,15)(12,16)(18,23)(21,28)(24,31)(26,32)(33,36,39,37)(34,41,38,45)(35,43,44,46)(40,47,42,48);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,19,22,27)(18,24,23,31)(20,29,30,25)(21,26,28,32)(33,35)(34,40)(36,43)(37,46)(38,42)(39,44)(41,47)(45,48);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29)(33,37,39,36)(34,41,38,45)(35,43,44,46)(40,48,42,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)(33,38,39,34)(35,42,44,40)(36,41,37,45)(43,47,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=g,f^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, 8800886);