#9783549

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	#45647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 9690745);