#8416745

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

3979 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8323941);