#8258588

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	#40175	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18)(19,24)(20,26)(21,28)(22,27)(23,25)(29,32)(30,31)(33,34,38,37)(35,39,41,45)(36,40,43,44)(42,46,47,48);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(17,19)(18,24)(20,22)(21,25)(23,28)(26,27)(29,30)(31,32)(35,41)(36,43)(39,45)(40,44);(2,8)(5,16)(7,11)(9,13)(18,25)(19,28)(22,29)(26,31)(33,35,38,41)(34,39,37,45)(36,42,43,47)(40,46,44,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,20)(18,26)(19,29)(21,27)(22,28)(23,30)(24,32)(25,31);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,21)(18,19)(20,27)(22,31)(23,24)(25,28)(26,29)(30,32)(33,36,38,43)(34,40,37,44)(35,42,41,47)(39,46,45,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,22,23,29)(18,27,25,32)(19,20,28,30)(21,31,24,26)(33,37,38,34)(35,39,41,45)(36,44,43,40)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8165784);