#9243832

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	#55939	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9151028);