#8517484

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	#38483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,2,6,7)(3,8,13,16)(4,9,14,12)(5,10,15,11)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(34,38)(35,42)(40,44)(43,48);(2,7)(4,14)(8,16)(10,11)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,8)(4,11)(5,9)(6,13)(7,16)(10,14)(12,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,9)(3,11)(5,8)(6,14)(7,12)(10,13)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5)(2,10)(3,12)(4,8)(6,15)(7,11)(9,13)(14,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8424680);