#8712345

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619541);