#7977521

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	#50835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7884717);