#8417217

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

3983 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,26,20,24)(21,25,29,30)(27,32,28,31)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,6)(3,9)(8,13)(11,15)(18,23)(24,26)(25,30)(31,32)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(4,12)(8,13)(11,15)(14,16)(17,19,22,20)(18,24,23,26)(21,28,29,27)(25,32,30,31)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,19)(18,24,23,26)(21,28,29,27)(25,31,30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,25,23,30)(19,27,20,28)(24,31,26,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gi,e^2=i,f^2=g,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, 8324413);