#8276812

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8184008);