#10407821

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

4304 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10315017);