#10038501

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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)(19,23)(20,30)(21,32)(22,24)(25,31)(26,28)(27,29)(33,34,38,43)(35,40,46,39)(36,41,42,37)(44,48,47,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,21)(22,27)(24,29)(25,26)(28,31)(30,32)(33,35,38,46)(34,39,43,40)(36,47,42,44)(37,45,41,48);(2,6)(7,10)(8,12)(14,16)(18,24)(19,27)(20,31)(26,32)(33,35,38,46)(34,39,43,40)(36,47,42,44)(37,45,41,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20)(18,25)(19,28)(21,27)(22,31)(23,32)(24,30)(26,29)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18)(19,29)(20,25)(21,32)(22,24)(23,27)(26,28)(30,31)(33,36)(34,41)(35,44)(37,43)(38,42)(39,45)(40,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,28)(18,26,24,32)(19,20,27,31)(23,25,29,30)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,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, 9945697);