#8322337

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

3847 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,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, 8229533);