#8322248

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	#42620	C₂
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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229444);