#7842410

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

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7749606);