#8928020

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835216);