#10037717

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3893 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^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, 9944913);