#8355726

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	#55462	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8262922);