#9727477

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32)(35,42)(37,39)(43,48)(45,47);(3,9)(7,13)(10,15)(14,16)(17,19,23,21)(18,24,27,28)(20,29,26,31)(22,30,25,32)(33,34,38,41)(35,39,42,37)(36,44,46,40)(43,47,48,45);(2,6)(7,13)(8,11)(14,16)(33,35,38,42)(34,37,41,39)(36,43,46,48)(40,47,44,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,26)(18,25,27,22)(19,29,21,31)(24,32,28,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,45)(44,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,19)(18,24,27,28)(20,31,26,29)(22,30,25,32)(33,36,38,46)(34,40,41,44)(35,43,42,48)(37,47,39,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,25)(18,26,27,20)(19,30,21,32)(24,31,28,29)(33,37,38,39)(34,35,41,42)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9634673);