#8393746

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

4123 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(2,6)(3,10)(8,13)(11,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(4,12)(8,13)(11,16)(14,15)(18,22)(24,25)(26,29)(31,32)(34,41)(35,43)(38,48)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,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, 8300942);