#8159091

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

3835 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=gh,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, 8066287);