#9685536

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(17,18,23,25)(19,28,22,24)(20,30,31,26)(21,27,32,29)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,30,32,26)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(18,25)(20,31)(24,28)(27,29)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,20,23,31)(18,26,25,30)(19,29,22,27)(21,28,32,24)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,5,7,2)(3,10,8,4)(6,9,14,13)(11,12,15,16)(17,21)(18,27)(19,26)(20,28)(22,30)(23,32)(24,31)(25,29)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,6,7,14)(2,9,5,13)(3,11,8,15)(4,12,10,16)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,26,32,30)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 9592732);