#9685584

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=gh,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, 9592780);