#7743629

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

4101 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)(11,15)(12,17)(13,18)(14,19)(16,20)(21,22)(23,24)(25,26,31,33)(27,29,28,35)(30,38,39,32)(34,40,36,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,27,31,28)(26,29,33,35)(30,34,39,36)(32,37,38,40);(11,20)(12,22)(15,16)(17,21)(25,28,31,27)(26,29,33,35)(30,36,39,34)(32,37,38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,38)(39,40);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,18)(13,15)(14,22)(16,23)(19,21)(20,24)(25,30)(26,32)(27,34)(28,36)(29,37)(31,39)(33,38)(35,40);(9,13,14,23)(10,18,19,24)(11,21,20,17)(12,15,22,16)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650825);