#8341621

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	#54289	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,23,21)(19,22,25,27)(20,24,29,30)(26,28,31,32)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(2,7)(3,12)(9,14)(10,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,28)(29,31)(30,32)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(2,7)(5,16)(8,11)(9,14)(19,25)(22,27)(26,31)(28,32)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,31)(27,32)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,18)(19,27,25,22)(20,24,29,30)(26,28,31,32)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,27)(18,25,21,19)(20,28,29,32)(24,31,30,26)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8248817);