#8700365

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(4,15)(5,10)(6,14)(7,12)(8,13)(11,16)(17,18,24,27)(19,28,30,22)(20,25,21,31)(23,26,29,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3,8,7)(2,9,13,12)(4,11,5,14)(6,15,16,10)(17,19)(18,22)(20,29)(21,23)(24,30)(25,26)(27,28)(31,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,4,8,5)(2,10,13,15)(3,14,7,11)(6,9,16,12)(17,20,24,21)(18,25,27,31)(19,29,30,23)(22,26,28,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,5,8,4)(2,10,13,15)(3,14,7,11)(6,12,16,9)(17,21,24,20)(18,25,27,31)(19,29,30,23)(22,32,28,26)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,6,8,16)(2,11,13,14)(3,15,7,10)(4,9,5,12)(17,22)(18,19)(20,26)(21,32)(23,31)(24,28)(25,29)(27,30)(33,36,38,35)(34,40,42,39)(37,43,46,44)(41,47,45,48);(1,7,8,3)(2,12,13,9)(4,11,5,14)(6,15,16,10)(17,23,24,29)(18,26,27,32)(19,21,30,20)(22,25,28,31)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607561);