#8418127

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3781 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(19,25)(22,29)(26,31)(30,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(3,12)(6,13)(8,16)(10,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gi,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, 8325323);