#8893417

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^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, 8800613);