#9863381

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	#49839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=hi,e^2=ghi,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, 9770577);