#8188197

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	#50051	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34)(35,42)(36,45)(37,46)(38,40)(39,41)(43,44)(47,48);(1,2,5,6)(3,7,10,12)(4,8,11,13)(9,14,15,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,39,38)(34,40,41,42)(36,43,46,48)(37,44,45,47);(4,11)(8,13)(9,15)(14,16)(18,24)(19,25)(26,29)(27,32)(33,36)(34,37)(35,43)(38,48)(39,46)(40,44)(41,45)(42,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37)(34,36)(35,44)(38,47)(39,45)(40,43)(41,46)(42,48);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,38,39,35)(34,40,41,42)(36,43,46,48)(37,47,45,44)

Polycyclic presentation:

2^9|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, 8095393);