#8818361

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	#38689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(34,38)(36,43)(39,44)(42,47);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(35,41)(39,44)(42,47)(46,48);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,23,22)(19,24,21,26)(20,28,30,25)(27,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22,23,18)(19,24,21,26)(20,28,30,25)(27,31,29,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|c^2=gi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8725557);