#10195904

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	#49083	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,31)(29,30)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,24)(20,29)(21,27)(22,30)(23,28)(25,31)(26,32)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(2,3)(7,9)(8,10)(14,15)(18,25)(19,28)(20,27)(26,30)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,30,31,26)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,29)(18,26,25,30)(19,20,28,27)(22,31,32,24)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,27)(19,30)(20,25)(21,24)(23,32)(26,28)(29,31)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,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, 10103100);