#8463187

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=h,e^2=hi,f^2=i,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, 8370383);