#10482510

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

4023 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)(3,8)(4,7)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(36,44)(39,45)(42,48)(46,47);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(18,24)(20,25)(26,31)(28,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(1,2,7,4)(3,10,6,8)(5,9,13,12)(11,16,14,15)(17,19,23,21)(18,20,24,25)(22,27,30,29)(26,28,31,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4,7,2)(3,8,6,10)(5,9,13,12)(11,16,14,15)(17,20,23,25)(18,19,24,21)(22,28,30,32)(26,27,31,29)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,19)(18,25,24,20)(22,27,30,29)(26,28,31,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6,7,3)(2,8,4,10)(5,11,13,14)(9,16,12,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=h,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, 10389706);