#8107666

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	#44152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,7)(4,10)(5,13)(6,9)(8,14)(11,16)(12,15)(18,23)(19,28)(20,30)(21,31)(34,40)(36,44)(38,42)(43,45);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34)(35,38)(36,41)(37,42)(39,40)(43,46)(44,47)(45,48);(1,2,6,9)(3,10,4,7)(5,8,14,13)(11,16,12,15)(17,19,22,28)(18,24,23,25)(20,27,30,32)(21,29,31,26)(33,35,39,37)(34,38,40,42)(36,45,44,43)(41,48,47,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19)(18,25)(20,27)(21,29)(22,28)(23,24)(26,31)(30,32)(33,36,39,44)(34,41,40,47)(35,43,37,45)(38,46,42,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20)(18,26)(19,27)(21,24)(22,30)(23,29)(25,31)(28,32)(33,37,39,35)(34,42,40,38)(36,45,44,43)(41,48,47,46);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,22,31)(18,27,23,32)(19,29,28,26)(20,24,30,25)(33,38,39,42)(34,35,40,37)(36,46,44,48)(41,43,47,45)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014862);