#8395004

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	#48551	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

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

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, 8302200);