#8893537

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 8800733);