#8927047

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

4524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,e^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, 8834243);