#8406865

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

3987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^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, 8314061);