#8418430

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

4121 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,21,24,26)(20,23,31,29)(27,30,28,32)(34,40)(35,37)(42,47)(43,45);(2,6)(3,9)(8,13)(11,15)(17,19)(18,21)(20,28)(22,24)(23,32)(25,26)(27,31)(29,30)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(4,12)(8,13)(11,15)(14,16)(19,24)(21,26)(27,28)(30,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,26)(20,29)(21,24)(22,25)(23,31)(27,30)(28,32)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325626);