#8323525

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	#52972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,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, 8230721);