#7901054

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

4127 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808250);