#9783307

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	#45596	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^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, 9690503);