#8118856

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	#46494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gi,f^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, 8026052);