#8709113

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	#38512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,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, 8616309);