#8118840

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	#52530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026036);