#8417278

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	#42409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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)(3,8)(4,6)(5,13)(7,9)(10,16)(11,12)(14,15)(18,22)(23,28)(24,29)(30,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,7,6)(2,8,9,4)(5,14,16,11)(10,12,13,15)(19,25)(23,28)(26,31)(30,32)(34,39)(35,42)(41,44)(43,48);(2,9)(3,6)(10,13)(11,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,3)(5,15)(6,9)(7,8)(10,14)(11,13)(12,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,7,16)(2,10,9,13)(3,11,6,14)(4,12,8,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,6,7,3)(2,4,9,8)(5,14,16,11)(10,12,13,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324474);