#7770283

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

4280 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)(3,7,8,5)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40);(11,17)(15,18)(19,23)(22,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7677479);