#10068975

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	#43278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=i,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, 9976171);