#8110623

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	#45593	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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:

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

Polycyclic presentation:

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

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, 8017819);