#8963316

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	#55519	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,22)(23,26)(24,29)(30,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(2,6)(4,13)(7,9)(10,16)(19,25)(20,28)(23,26)(24,29)(35,41)(37,45)(39,42)(40,44);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,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=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870512);