#8284447

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

4247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191643);