#8445999

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

4022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353195);