#10480849

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

4397 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32)(36,44)(39,45)(42,48)(46,47);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,20)(18,24)(19,21)(22,29)(23,25)(26,31)(27,32)(28,30)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gi,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, 10388045);