#10442529

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3864 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)(3,8,6,4)(5,13,15,9)(11,12,14,16)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32)(33,34,39,37)(35,40,46,45)(36,42,47,41)(38,48,44,43);(1,3,7,6)(2,8,10,4)(5,14,15,11)(9,12,13,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35,39,46)(34,40,37,45)(36,44,47,38)(41,48,42,43);(1,4)(2,6)(3,10)(5,12)(7,8)(9,14)(11,13)(15,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,30,32,27)(33,36)(34,41)(35,44)(37,42)(38,46)(39,47)(40,48)(43,45);(1,4)(2,3)(5,12)(6,10)(7,8)(9,11)(13,14)(15,16)(17,19)(18,20)(21,26)(22,30)(23,24)(25,29)(27,31)(28,32)(33,37,39,34)(35,45,46,40)(36,42,47,41)(38,48,44,43);(1,5,7,15)(2,9,10,13)(3,11,6,14)(4,12,8,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,30,32,27)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,43)(45,48);(1,6,7,3)(2,8,10,4)(5,14,15,11)(9,16,13,12)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,38,39,44)(34,43,37,48)(35,36,46,47)(40,41,45,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=hi,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, 10349725);