#10455715

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

4130 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)(20,28)(23,29)(26,32)(30,31)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,35)(34,39)(36,47)(37,44)(38,45)(40,48)(41,46)(42,43);(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,36,38,44)(34,40,42,46)(35,37,45,47)(39,41,43,48);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20)(18,23)(19,26)(21,28)(22,30)(24,31)(25,29)(27,32)(33,35)(34,39)(36,37)(38,45)(40,41)(42,43)(44,47)(46,48);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gh,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, 10362911);