#9685566

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,8)(4,12)(5,9)(6,10)(7,14)(11,13)(15,16)(17,18,23,25)(19,28,22,24)(20,21,27,26)(29,30,31,32)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3)(2,5)(4,11)(6,9)(7,13)(8,10)(12,15)(14,16)(17,19,23,22)(18,24,25,28)(20,29,27,31)(21,32,26,30)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(18,25)(20,27)(24,28)(29,31)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,6,14)(2,7,10,12)(3,11,9,16)(5,13,8,15)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,30,28,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,15)(12,13)(14,16)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,29,28,31)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5,6,8)(2,9,10,3)(4,13,14,15)(7,16,12,11)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,30,26,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9592762);