#8228521

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

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

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, 8135717);