#8741967

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	#52695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649163);