#8810201

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,26,21,24)(20,30,31,25)(22,27,29,32)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(2,6)(7,9)(8,13)(14,15)(17,19,23,21)(18,24,28,26)(20,22,31,29)(25,27,30,32)(37,45)(41,44)(43,46)(47,48);(3,10)(4,12)(7,9)(8,13)(17,20,23,31)(18,25,28,30)(19,29,21,22)(24,32,26,27)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,23,19)(18,26,28,24)(20,22,31,29)(25,27,30,32)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,22)(18,27)(19,20)(21,31)(23,29)(24,25)(26,30)(28,32)(33,37,39,45)(34,41,36,44)(35,43,38,46)(40,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)

Polycyclic presentation:

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

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, 8717397);