#8393613

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	#49083	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,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, 8300809);