#8504794

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3742 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411990);