#7711073

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

4290 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)(11,16)(12,17)(13,15)(14,21)(18,22)(19,24)(20,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27,31,30)(26,32,33,35)(28,36,34,38)(29,37,40,39);(1,3,5,4)(2,7,6,8)(10,13)(11,12)(18,21)(19,20)(26,33)(28,34)(32,35)(36,38);(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28,31,34)(26,29,33,40)(27,36,30,38)(32,37,35,39);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,29)(26,34)(27,37)(28,33)(30,39)(31,40)(32,38)(35,36);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,30,31,27)(26,35,33,32)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=i,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, 7618269);