#8180381

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	#54388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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)(3,9)(4,5)(6,10)(7,8)(11,16)(12,13)(14,15)(17,18)(19,27)(20,31)(21,32)(22,26)(23,24)(25,28)(29,30)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(2,8)(4,9)(10,15)(12,16)(18,24)(19,28)(20,32)(26,30)(36,44)(40,46)(43,47)(45,48);(1,3,7,5)(2,9,8,4)(6,13,14,11)(10,12,15,16)(17,19,23,28)(18,25,24,27)(20,29,32,22)(21,30,31,26)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,15)(14,16)(17,20)(18,21)(19,29)(22,28)(23,32)(24,31)(25,30)(26,27)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,5,7,3)(2,4,8,9)(6,13,14,11)(10,12,15,16)(17,21)(18,20)(19,30)(22,27)(23,31)(24,32)(25,29)(26,28)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,29)(18,26,24,30)(19,20,28,32)(21,27,31,25)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087577);