#9719164

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	#49683	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,32)(29,31)(33,34,40,42)(35,43,37,41)(36,46,38,39)(44,45,47,48);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27)(33,35,40,37)(34,41,42,43)(36,44,38,47)(39,45,46,48);(2,8)(3,11)(4,14)(10,15)(17,20,22,21)(18,24,27,25)(19,28,30,29)(23,31,26,32)(33,36,40,38)(34,39,42,46)(35,44,37,47)(41,45,43,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,21,22,20)(18,25,27,24)(19,29,30,28)(23,32,26,31)(33,37,40,35)(34,41,42,43)(36,44,38,47)(39,48,46,45);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,38,40,36)(34,39,42,46)(35,44,37,47)(41,48,43,45);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,39)(34,38)(35,45)(36,42)(37,48)(40,46)(41,47)(43,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=i,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, 9626360);