#8517589

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=g,f^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, 8424785);