#7709724

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	C₂×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616920);