#9814616

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 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)(3,12)(4,6)(5,8)(7,9)(10,11)(13,16)(14,15)(17,18,23,21)(19,27,29,24)(20,30,28,25)(22,26,32,31)(33,34)(35,37)(36,41)(38,47)(39,40)(42,44)(43,46)(45,48);(1,3,7,5)(2,8,9,12)(4,13,11,15)(6,14,10,16)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31)(33,35)(34,37)(36,44)(38,45)(39,43)(40,46)(41,42)(47,48);(2,9)(4,11)(8,12)(13,15)(34,40)(35,43)(36,45)(42,47);(1,4,7,11)(2,10,9,6)(3,13,5,15)(8,16,12,14)(17,20,23,28)(18,25,21,30)(19,22,29,32)(24,26,27,31)(33,36,39,45)(34,41,40,48)(35,44,43,38)(37,42,46,47);(1,5,7,3)(2,8,9,12)(4,15,11,13)(6,14,10,16)(17,21,23,18)(19,24,29,27)(20,30,28,25)(22,26,32,31)(33,37)(34,35)(36,42)(38,48)(39,46)(40,43)(41,44)(45,47);(1,6,7,10)(2,11,9,4)(3,14,5,16)(8,15,12,13)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)(33,38,39,44)(34,42,40,47)(35,45,43,36)(37,41,46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9721812);