#8925139

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

4005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,f^2=g,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, 8832335);