#10042582

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9949778);