#7986329

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893525);