#8291662

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	#55261	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(34,40)(35,42)(41,44)(43,48);(2,9)(6,10)(8,11)(13,16)(19,25)(23,26)(27,31)(30,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,13)(11,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,3)(2,11,9,8)(4,12,15,14)(6,16,10,13)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=g,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, 8198858);