#8718412

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(33,34,39,43)(35,44,46,40)(36,41,45,37)(38,42,47,48);(2,7)(3,12)(9,14)(10,13)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(33,35,39,46)(34,40,43,44)(36,38,45,47)(37,42,41,48);(2,7)(3,12)(9,14)(10,13)(19,28)(20,29)(24,31)(25,30)(33,36,39,45)(34,41,43,37)(35,38,46,47)(40,48,44,42);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,16)(10,15)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,36)(34,37)(35,38)(39,45)(40,42)(41,43)(44,48)(46,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,37)(34,36)(35,42)(38,40)(39,41)(43,45)(44,47)(46,48);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,38)(34,42)(35,45)(36,46)(37,44)(39,47)(40,41)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625608);