#8271320

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	#53171	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=i,f^2=ghi,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, 8178516);