#8436627

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,24,28)(19,29,23,25)(20,31,30,26)(21,32,27,22)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19,24,23)(18,25,28,29)(20,27,30,21)(22,31,32,26)(34,39)(35,40)(41,46)(42,47);(17,20)(18,26)(19,21)(22,29)(23,27)(24,30)(25,32)(28,31)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,21,24,27)(18,22,28,32)(19,30,23,20)(25,26,29,31)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,22)(18,27)(19,31)(20,29)(21,28)(23,26)(24,32)(25,30);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,23,24,19)(18,25,28,29)(20,27,30,21)(22,26,32,31)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343823);