#10529828

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3777 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)(17,18,22,25)(19,21,24,27)(20,23,30,32)(26,29,28,31)(33,34,39,37)(35,42,38,40)(36,41,45,46)(43,48,44,47);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19)(18,21)(20,28)(22,24)(23,31)(25,27)(26,30)(29,32);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,32)(27,31)(33,35,39,38)(34,40,37,42)(36,43,45,44)(41,47,46,48);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,18)(19,27)(20,23)(21,24)(22,25)(26,31)(28,29)(30,32)(33,36)(34,41)(35,43)(37,46)(38,44)(39,45)(40,47)(42,48);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,20)(18,23)(19,28)(21,31)(22,30)(24,26)(25,32)(27,29)(33,37,39,34)(35,42,38,40)(36,41,45,46)(43,47,44,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,27)(18,24,25,19)(20,29,30,31)(23,28,32,26)(33,38,39,35)(34,42,37,40)(36,44,45,43)(41,48,46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 10437024);