#10201962

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	#41662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10109158);