#9713722

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	#38317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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)(3,12)(4,10)(5,11)(6,8)(7,9)(13,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,30)(26,32)(27,28)(29,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,7,6)(2,8,9,12)(4,13,11,15)(5,16,10,14)(17,19)(18,24)(20,28)(21,30)(22,25)(23,27)(26,31)(29,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,9)(4,11)(8,12)(13,15)(18,20)(19,27)(25,29)(26,30)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,7,11)(2,10,9,5)(3,13,6,15)(8,14,12,16)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,5,7,10)(2,11,9,4)(3,14,6,16)(8,13,12,15)(17,21,23,31)(18,25,20,29)(19,26,27,30)(22,24,32,28)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,6,7,3)(2,12,9,8)(4,15,11,13)(5,14,10,16)(17,22)(18,26)(19,25)(20,30)(21,28)(23,32)(24,31)(27,29)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=g,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, 9620918);