#7888124

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

4217 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)(9,10,16,21)(11,15,20,24)(12,19,14,17)(13,18,22,23);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,22)(16,20)(17,23)(18,19)(21,24)(26,30)(31,37)(32,38)(39,40);(1,3,5,4)(2,7,6,8)(9,12,16,14)(10,17,21,19)(11,22,20,13)(15,18,24,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,13,16,22)(10,18,21,23)(11,12,20,14)(15,19,24,17);(1,4,5,3)(2,8,6,7)(9,14,16,12)(10,19,21,17)(11,22,20,13)(15,18,24,23)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,15,16,24)(10,20,21,11)(12,18,14,23)(13,19,22,17)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7795320);