#8188227

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	#40822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

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

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, 8095423);