#9815370

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	#55042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,e^2=g,f^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, 9722566);