#8824769

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

4299 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)(3,10,8,4)(5,13,14,9)(11,12,15,16)(17,18)(19,26)(20,21)(22,23)(24,27)(25,31)(28,29)(30,32)(35,41)(39,42)(43,47)(46,48);(2,7)(4,10)(5,14)(11,15)(18,23)(21,25)(24,26)(29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,7)(3,8)(9,13)(11,15)(19,27)(24,26)(28,30)(29,32)(34,38)(39,42)(40,44)(46,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,22,27)(18,24,23,26)(20,30,31,28)(21,29,25,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,21,22,25)(18,20,23,31)(19,29,27,32)(24,30,26,28)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731965);