#8197246

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	#56698	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104442);