#9685587

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,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, 9592783);