#8741915

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(2,7)(3,11)(9,12)(10,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(34,40)(35,36)(42,45)(43,44);(2,7)(3,11)(9,12)(10,16)(20,28)(24,31)(27,29)(30,32)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8649111);