#8167398

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

3940 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074594);