#8158722

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	#52365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,39)(35,42)(36,44)(45,47);(2,7)(3,9)(10,13)(11,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(2,7)(3,9)(10,13)(11,16)(20,29)(24,28)(26,32)(30,31)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065918);