#9863039

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	#48318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,24)(19,26,21,23)(20,25,28,31)(27,32,29,30)(34,40)(36,46)(41,47)(43,44);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,21)(18,23,24,26)(20,27,28,29)(25,30,31,32)(33,34,39,40)(35,41,37,47)(36,42,46,45)(38,43,48,44);(2,3)(7,9)(8,10)(14,15)(18,24)(23,26)(25,31)(30,32)(33,35,39,37)(34,41,40,47)(36,44,46,43)(38,45,48,42);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,36)(34,42)(35,44)(37,43)(38,41)(39,46)(40,45)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,37,39,35)(34,41,40,47)(36,43,46,44)(38,45,48,42);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,19)(18,26,24,23)(20,27,28,29)(25,30,31,32)(33,38)(34,43)(35,45)(36,41)(37,42)(39,48)(40,44)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=i,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, 9770235);