#8761995

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	#48493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8669191);