#8926474

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3715 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=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, 8833670);