#9717387

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	#50034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,8,6,16)(4,10,13,5)(11,12,14,15)(17,18,22,24)(19,23,27,30)(20,25,29,32)(21,28,31,26)(33,34)(35,44)(36,39)(37,42)(38,41)(40,43)(45,48)(46,47);(1,3,7,6)(2,8,9,16)(4,11,13,14)(5,15,10,12)(17,19,22,27)(18,23,24,30)(20,28,29,26)(21,25,31,32)(33,35,40,38)(34,41,43,44)(36,45,37,46)(39,47,42,48);(2,9)(4,13)(8,16)(11,14)(18,24)(19,27)(25,32)(26,28)(33,36,40,37)(34,39,43,42)(35,45,38,46)(41,47,44,48);(1,4)(2,10)(3,11)(5,9)(6,14)(7,13)(8,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,37,40,36)(34,42,43,39)(35,46,38,45)(41,48,44,47);(1,5,7,10)(2,4,9,13)(3,12,6,15)(8,14,16,11)(17,20,22,29)(18,25,24,32)(19,26,27,28)(21,30,31,23)(33,38,40,35)(34,41,43,44)(36,46,37,45)(39,47,42,48);(1,6,7,3)(2,8,9,16)(4,14,13,11)(5,15,10,12)(17,21)(18,26)(19,25)(20,23)(22,31)(24,28)(27,32)(29,30)(33,39,40,42)(34,36,43,37)(35,47,38,48)(41,45,44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9624583);