#7972724

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=g,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, 7879920);