#8933869

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	#50784	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841065);