#8107274

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	#52963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014470);