#10480991

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

4010 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:

(1,2,7,10)(3,11,5,6)(4,9,15,8)(12,14,13,16)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(36,44)(39,45)(42,48)(46,47);(1,3,7,5)(2,6,10,11)(4,12,15,13)(8,14,9,16)(18,24)(20,25)(26,31)(28,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(1,4)(2,8)(3,12)(5,13)(6,14)(7,15)(9,10)(11,16)(17,19,23,21)(18,25,24,20)(22,27,30,29)(26,32,31,28)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4)(2,9)(3,13)(5,12)(6,14)(7,15)(8,10)(11,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,5,7,3)(2,6,10,11)(4,13,15,12)(8,14,9,16)(17,21,23,19)(18,20,24,25)(22,27,30,29)(26,32,31,28)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6)(2,5)(3,10)(4,14)(7,11)(8,13)(9,12)(15,16)(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|a^2=g,b^2=h,c^2=gh,d^2=i,e^2=g,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, 10388187);