#8374197

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8281393);