#8990318

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	#43418	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8897514);