#8393397

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,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, 8300593);