#10519367

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

4220 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34)(35,39)(36,46)(37,41)(38,42)(40,45)(43,48)(44,47);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(35,43)(36,45)(39,48)(40,46);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35)(34,39)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36,38,45)(34,40,42,46)(35,44,43,37)(39,41,48,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,38,44)(34,41,42,47)(35,45,43,36)(39,40,48,46);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,44)(43,48)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10426563);