#10529204

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3784 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)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18,22,25)(19,21,24,27)(20,23,30,32)(26,29,28,31)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(2,7)(3,10)(9,15)(11,16)(17,19)(18,21)(20,28)(22,24)(23,31)(25,27)(26,30)(29,32)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,32)(27,31)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,18)(19,27)(20,23)(21,24)(22,25)(26,31)(28,29)(30,32)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,20)(18,23)(19,28)(21,31)(22,30)(24,26)(25,32)(27,29)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,21,22,27)(18,24,25,19)(20,29,30,31)(23,28,32,26)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,f^2=gi,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, 10436400);