#10442524

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,8,4,5)(6,9,14,16)(11,15,12,13)(17,18)(19,24)(20,26)(21,22)(23,27)(25,31)(28,30)(29,32)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,19,23,22)(18,24,27,21)(20,29,31,28)(25,30,26,32)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(1,3,7,4)(2,8,10,5)(6,11,14,12)(9,15,16,13)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,32)(26,27)(33,36)(34,41)(35,44)(37,43)(38,45)(39,46)(40,47)(42,48);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,20)(18,26)(19,29)(21,30)(22,28)(23,31)(24,32)(25,27)(33,37,39,41)(34,36,43,46)(35,42,45,47)(38,40,44,48);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,21,23,24)(18,19,27,22)(20,30,31,32)(25,28,26,29)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,19)(18,24,27,21)(20,29,31,28)(25,32,26,30)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10349720);