#8734644

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

4024 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(34,38)(36,43)(39,44)(42,47);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(35,41)(39,44)(42,47)(46,48);(3,11)(4,12)(7,16)(8,13)(18,20)(24,27)(25,30)(26,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20,23,18)(19,24,29,27)(21,30,28,25)(22,26,32,31)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641840);