#7709807

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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)(9,10,15,17)(11,16,14,12)(13,21,23,18)(19,20,22,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,17,12)(13,19,23,22)(18,24,21,20)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(10,17)(12,16)(18,21)(20,24)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(9,12)(10,11)(13,20)(14,17)(15,16)(18,19)(21,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,13,15,23)(10,18,17,21)(11,19,14,22)(12,20,16,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(9,14,15,11)(10,12,17,16)(13,22,23,19)(18,20,21,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,e^2=gh,f^2=h,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, 7617003);